Galzolari E., Filippucci R., Pucci P. - Existence of radial solutions for the p-laplacian elliptic...

7/25/2019 Galzolari E., Filippucci R., Pucci P. - Existence of radial solutions for the p-laplacian elliptic equations with weights(

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Manuscirpt submitted to Website: http://AIMsciences.orgAIMS JournalsVolume 15, Number 2, June 2006 pp. 447479

EXISTENCE OF RADIAL SOLUTIONS

FOR THE PLAPLACIAN ELLIPTIC EQUATIONS WITH WEIGHTS

Elisa Calzolari, Roberta Filippucci and Patrizia Pucci

Dipartimento di Matematica e InformaticaUniversita degli Studi di Perugia

Via Vanvitelli 106123 Perugia, Italy

(Communicated by Eiji Yanagida)

Abstract. Using the definition of solution and the qualitative properties es-tablished in the recent paper [17], some existence results are obtained both forcrossing radial solutions and for positive or compactly supported radial groundstates in Rn of quasilinear singular or degenerate elliptic equations with weightsand with nonlinearities which can be possibly singular at x = 0 and u = 0,respectively. The technique used is based on the papers [1] and [12]. Fur-thermore we obtain a nonexistence theorem for radial ground states using atechnique of Ni and Serrin [13].

1. Introduction. Recently, in [17] for the pLaplacian equation with weights inRn, under general conditions for the nonlinearity f, uniqueness of ground states

and various qualitative properties of solutions were established. Here we prove

existence of crossing radial solutions for f positive near u = 0, and existence ofradial ground states in Rn for f negative near u = 0 of such spatially dependentequations. More specifically, we use a unified proof and a new subcritical condition() on fat infinity, which was introduced in [1]. Indeed, in canonical cases, () isinteresting in applications and () is more general than the well known subcriticalcondition (1) of Castro and Kurepa [4], adopted in several related papers, as [22],[8] and [12].

In particular, we are interested in finding sufficient conditions for existence ofradial ground states of the singular quasilinear elliptic equation with weights

div(g(|x|)|Du|p2Du) + h(|x|)f(u) = 0 in Rn \ {0},

p >1, n1, (1)

whereg, h:R+

R+

and Du= (u/x1, , u/xn), when f 1, (2)

2000 Mathematics Subject Classification. Primary: 35J70; Secondary: 35J60.Key words and phrases. Ground states, pLaplacian operator, weight functions.This research was supported by the Italian MIUR project titled Metodi Variazionali ed

Equazioni Differenziali non Lineari.

447

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448 E. CALZOLARI, R. FILIPPUCCI AND P. PUCCI

wherea(r) =rn1g(r), b(r) = rn1h(r), n 1 and r = |x|. The simple LaplacePoisson equation arises when a(r) =b(r) = rn1, where n is the underlying spacedimension.

Moreover, with the same technique, when f >0 near u = 0, we are also able toprove the existence of a radial crossing solution of (1) in its maximal continuationinterval where u > 0 and u < 0. We recall that these existence results wereestablished in [1] for general quasilinear elliptic equations without weights, while in[22] for the pLaplacian equation without weights.

In addition to the ground state problem, when f < 0 near u = 0, we can alsoconsider existence of nontrivial radial solutions of the homogeneous DirichletNeumann free boundary problem

div(g(|x|)|Du|p2Du) + h(|x|)f(u) = 0 in BR\ {0},

u 0, u0, u= u= 0 on BR,(3)

for someR >0.A number of examples fall into the general category of (1). A first is the ce-

lebrated Matukuma equation and several generalizations of it in stellar dynamics,cfr. [11], [2], [23], [5], [10], [7] and [14][17]. All these models are discussed in detailin Section 4 of [17], as special cases of the main example introduced in [17]

div(rk|Du|p2Du) + r

rs

1 + rs

/sf(u) = 0,

p >1, n 1, k R, R, s > 0, >0.

(4)

In particular, under the following general conditions on the exponents

k p, k

p

+

p 1 n, (5)

wherep is the Holder conjugate ofp, and on f

(f1) fC(R+) L1[0, 1],

in [17] a careful definition ofsemiclassical solution for (1) was given, as well as aqualitative theory. Finally the main uniqueness theorem of [17] can be applied to(4) under appropriate behavior of the nonlinearity f(u), satisfied i.e. by

f(u) = um + u; p 2, 1< m < p 1, m 1 +p 3

p 1. (6)

In other words radial nonnegative nonsingular semiclassical ground states for

equations of type (4)(6) are unique.Some existence and nonexistence results for radial ground states of special casesof (2) are given in [5] whenfis continuous also atu = 0 and nonnegative foru >0small. In the recent paper [9] some existence, nonexistence and uniqueness resultsfor radial ground states of some special cases of (2) are given when f >0 everywherein R+ but singular at u = 0. For a more detailed discussion and comparison withour results we refer to the Remarks after Theorems 5 and 6 in Section 7. Weemphasize, however, that the main case treated in the present paper is when f 0 small, as in [17].

Throughout the paper we shall adopt the definition of semiclassical solutionfor (1) proposed in [17], when f satisfies (f1) and F(u) =

u0

f(v)dv denotes the

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pLAPLACIAN ELLIPTIC EQUATIONS WITH WEIGHTS 449

well defined integral function off. In the main existence theorems of Section 7 wesuppose also that either

(f2) there exists >0 such thatF(u)< 0 for0 0,

as in [8], [12] and [1], or

(f3) there existsc > 0, possibly infinite, such thatf(u)> 0 for0< u < c,

as in [22] and [1].

Using the main change of variable of [17], we transform (2) into the equivalentequation

[q(t)|vt|p2vt]t+ q(t)f(v) = 0, (7)

that is, into an equation of type (2), but with the same weights. In the specialcase when (7) arises with q(t) = tN1 for some N 1, then earlier theory can beapplied, see e.g. [18] and [19], but of course, in general qis no longer a pure power.In order to study the existence of semiclassical solutions of (2), we ask that thetransformed equation (7) is compatible with the basic structureassumptions of [17],namely:

(q1) q C1(R+), q >0, qt> 0 in R+;

(q2) qt/q is strictly decreasing on R+;

(q3) limt0+

tqt(t)

q(t) =N 1 0.

The paper is organized as follows: in Section 2 the definition of semiclassical

solutions of (1) and preliminary qualitative properties for such solutions are given.In Section 3 we present and summarize the main properties of solutions of thecorresponding initial value problem, in the spirit of [6]. Section 4 is devoted toshowing the connections between the following subcritical growth conditions (1)

and (), with Q(t) =t

0q()d, d= under (f2) and d = 0 under (f3).

(1) The function(v) = pN F(v) (Np)vf(v), v R+, is locally bounded nearv = 0 and there exist > d and (0, 1) such that (v) 0 for all v and

lim supv

(1v)Q

C

vp1

f(2v)

1/p= for all 1, 2 [, 1],

whereC = [(1 )p]1/p

.Property (1) is equivalent to the famous condition of Castro and Kurepa [4], usedin [22], [8], [12] and [1] in the standard case, when q(t) = tN1. While property

() The function(v) = pN F(v) (Np)vf(v), v R+, is locally bounded nearv = 0 and there exist > d and (0, 1) such that (v) 0 for all v and

lim supv

(1v)Q

c[vp+1f(2v)]

1/p

= for every 1, 2 [, 1],

wherec= [(1 )p]1/p

(1 )2/p,

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is equivalent, when q(t) = tN1, to the condition introduced in [1]. Of coursec < C.

Section 4 ends with some remarks about the independence of the two differentgrowth hypotheses () and (1). Proposition 1 shows that, under(f1)and the as-sumptionlim infu f(u) = k0> 0, withk0 possibly, condition (1) is strongerthan (). Examples illustrating the independence of () and (1) are given inSection 5. For instance, when 1 >1, with 1< < pN, where pN =N p/(N p), see

Section 5 and also [1]. In Section 6 some preliminary lemmas are presented to sim-plify the main proofs. In Section 7 existence of crossing solutions is established when(f3) holds, as well as the principal existence theorems if (f2) is verified. Finally,in Section 8 a nonexistence theorem for positive radial semiclassical nonsingularground states of (1) is given under condition (f2).

From the main results of Sections 78, the following consequence can be derived.Corollary 1. Suppose that1< p < N. Assume(q1)(q3) and let

f(u) = um + u, 1< m < . (8)

(i)There exists a semiclassical nonsingular radial ground stateu of (1)when-ever

0< m < < pN 1,

and

(q4) limt

q(t)q(t)

= 0,

holds. Moreover, u is positive in the entireRn if and only ifm p 1, while it iscompactly supported when0< m < p 1.

(ii)There exists a positive radial semiclassical nonsingular solution of the cor-responding homogeneous DirichletNeumann free boundary value problem (3) if

1< m 0

is satisfied.Moreover, in cases (i) and (ii), if for some 0,

then the constructed solution u of (1) and (8) is regular, that is Du is Holdercontinuous atx = 0, withDu(0) =0; while if [1, p), thenu is Holder continuousatx= 0. In both cases

u W1,ploc(Rn),

when also 1< p n.

Conditions (q1)(q5) are given in terms of the main radial weights a and b in thepaper, cfr. (A1)(A4) in Section 2, (A5) in Section 3 and finally (A6) in Section 8as well as their related comments.


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2. Semi-classical solutions. Consider the quasilinear singular elliptic equation

div(g(|x|)|Du|p

2

Du) + h(|x|)f(u) = 0in ={x Rn \ {0} : u(x)> 0}; p >1, n 1,

u 0, u0 in Rn \ {0},

(9)

whereg, h: R+ R+. Prototypes of (9), with nontrivial functionsg,h, are given,for example, by equations of Matukuma type and equations of BattFaltenbacherHorst type, see (1.4) and Section 4 of [17]. In several interesting cases g can besingular at the origin, and in general h also may be singular there; thus it is necessaryin (9) that excludes the point x = 0 and also points where u(x) = 0 because ofthe assumption (f1) which allows fto be singular at u = 0.

We shall be interested in the radial version of (9), namely

[a(r)|u|p2u]+ b(r)f(u) = 0

in J={r R+ : u(r)> 0}, p >1, r= |x|,

u= u(r), u 0, u0 in R+,

(10)

where, with obvious notation,

a(r) =rn1g(r), b(r) = rn1h(r). (11)

As in [17], in order that the transformed equation (7) should satisfy the requirements(q1)(q3) we shall ask that the coefficients a, b have the following behavior

(A1) a,b >0 in R+, a, bC1(R+),

(A2) (b/a)1/p L1[0, 1] \ L1[1, ),

(A3) the function

(r) =

1

p

a

a +

1

p

b

b

ab

1/p

is positive and strictly decreasing inR+, wherep is the Holder conjugate ofp (>1),

(A4) there isN1 such that

limr0+

(r)

r0

ba

1/p=N 1.

In Section 4 of [17] several equations, such as (4), satisfying the above conditions

and modelling physical phenomena, are presented. As noted in [17], in the specialcase when g 1 or equivalently a(r) = rn1, assumptions (A1) and (A2) alsoappear in [10], though in somewhat different circumstances, see also [15] and [16].

As noted above, since (9) is possibly singular when x= 0 and when u= 0, it isnecessary to define carefully the meaning to be assigned to solutions of (9), and inturn, of (10). One can consider weak distribution solutions of (9), or alternativelydistribution solutions with suitable further regularity conditions and well definedvalues at x = 0. Following [17] we shall thus consider the following definition.

Definition. A semiclassical radial solution of (9) is a nonnegative function uof class C1(R+), which is a distribution solution of (10) in J, that is for all C1

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functions = (r), having compact support in J, it results

J

rn1g(r)|u|p2u dr =J

rn1h(r)f(u)dr.

In Proposition 2.1 of [17] it is proved that every semiclassical radial solution be-comes a classical solution in R+ whenf is continuous in R+0 with f(0) = 0. Condi-tions which guarantee nonsingular behavior of solutions of (10) at r = 0 are alsogiven in [17].

As noted in the Introduction, equation (2) can be transformed in equation (7)by the following change of variables of [17]

t(r) =

r0

[b(s)/a(s)]1/p ds, r 0. (12)

Of course t : R+0 R+0, t(0) = 0, t() = , by (A2), and t is a diffeomorphism of

R+

0

into R+

0

by (A1), with inverser = r(t),t 0. The relation between the originalweightsa and band the new weight qis given by

q(t) = [a(r(t))]1/p[b(r(t))]1/p

, t >0. (13)

Obviously, ifu = u(r) is a semiclassical solution of (10) in J, then v =v(t) =u(r(t)) is of class C1(R+) and it satisfies (7) in I = {t R+ : v(t) > 0}, namelyv is a semiclassical solution of (7). We emphasize that q C(R+0) C

1(R+) bycondition (q1). In particularv satisfies (7) in Iin the classical sense with v 0 and

v C1(R+0), |vt|p2vt C1(I). (14)

For details see Proposition 3.1 and Theorem 3.2 of [17] together with other relatedresults contained in Section 3 of [17].

In the main example (4) we have

a(r) = rn+k1, b(r) = rn+1

rs

1 + rs

/s

.

As shown in Section 4 of [17] conditions (A1)(A4) are satisfied if (5) is verified.Furthermore,

(r) =

n 1 +

k

p+

p+

p

1

1 + rs

1 + rs

rs

/ps r(k)/p1, (15)

and the limit value N 1 in (A4) is given byn 1 +

k

p+

p+

p

limr0+

r(k)/p1 r

0

t(k+)/pdt, (16)

which immediately yields

N=p n + +

p + + k >1, (17)

by (5).Finally 1 p n. The latter condition implies that

a1/(p1) L1[0, 1]. In this case, as proved in [17] via the main result of [7], thenatural Sobolev exponentof (4) and its transformed equation (7) is given by

1

pN=

1

p

1

N =

1

p

n + k p

n + + .

Here pN> p because N >1.

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To obtain the asymptotic behavior ofu(r) forrnear 0 one can apply Theorem 3.2of [17], since

g(r)

h(r)=

a(r)

b(r) r as r 0+, = k ,

and in turn (17) can be rewritten in the form

N=pn + k

p .

Thus from (3.10) of [17] we find as r 0+

r(1)/(p1)u(r) [sgn f()]

|f()|

n + k

1/(p1). (18)

It is finally interesting in this example that the parameter sin (4) does not appearin any of the exponent relations (5), (17)(18). This is a reflection of the fact thatthe term rs/(1 + rs) in (4) can be replaced by more general functions having thesame asymptotic behavior.

Conditions (5) have the first consequence that n. Moreover, eithera canbe discontinuous (k p n).

For example a is discontinuous ifn= 3, p = 2, k =5/2, = 1/2, while b isdiscontinuous andN > p whenn = 3,p = 2,k = 1/2, = 9/4 and 0< 1.

This is clear in the main example of [18], see also [17], when in (9)

g(r) 1, h(r) = r.

Indeed, here = 0 and (A1)(A4) hold if

+p >0, + (n 1)p > 0, (19)

with

N=p + n

+p >1. (20)

At the same time using the main change of variable (12) we see that (7) takes thecanonical form

[tN1|vt|p2vt]t+ tN1f(v) = 0,

by (12) and (13), that is, Nserves as the natural dimension for this example.

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For this case we have n > p if and only ifN > p. This condition also impliesthata1/(p1) =r(n1)/(p1) L1[0, 1]. Thus the natural Sobolev exponent is

1

pN=

1

p

1

N =

n p

+ n,

confirming again the role ofNas the natural dimension of the problem.For other equations modelled by (4) we refer to Section 4 of [17].

3. Preliminary Results. For simplicity, from this point on we write = d/dt ifthere is no confusion in the notation. In this section, as in [1], we present somepreliminary results useful for the proof of the main existence theorems of crossingsolutions and of radial ground states of (2) via equation (7). In particular, they aresemiclassical solutions of the initial value problem

[q(t)|v(t)|p2v(t)]+ q(t)f(v) = 0, t >0,v(0) =, v(0) = 0. (21)

Define

d:=

, if (f2) holds,

0, if (f3) holds.and := sup{v > d : f(u)> 0 for u(d, v)}.

Moreover, as in [1], from now on we assume together with (f1) also

(f4) fLiploc(0, ).

Finally, we restrict our attention to solutions v of (21), with

(d, ). (22)

Lemma 1. Assume that f satisfies either (f2) or (f3). If v is a semiclassicalsolution of (21) and (22), then v(t) < 0 near the origin. Moreover v is unique

until it exists and remains in(0, ), provided thatv(t)< 0.

Proof. From (22), we deduce that there exists t0> 0 sufficiently small such that

[q(t)|v(t)|p2v(t)] = q(t)f(v)< 0, t(0, t0).

Henceq(t)|v(t)|p2v(t) is strictly decreasing in (0, t0) and it assumes value zero att= 0 from (q1). Consequentlyv (t)< 0, t (0, t0). Thusv is a solution of the firstorder differential system

w(t) =

q(t)q(t)

w(t) f(v)

v(t) = [w(t)]1/(p1)

v(0) =, w(0) = 0,

(23)

where we have used the fact that sgn w(t) = sgn v (t), beingw(t) = |v(t)|p2v(t).Finally, (q1) and (f4) guarantee that (23) admits a unique solution such that

v(t)> 0 and < v(t)< 0

for all t >0.

Lemma 1 says that the unique local solution vof (21) and (22) can be continuedexactly until t , where t is the first point in R+, uniquely determined, suchthat eitherv(t) = 0 and v

(t) 0, or v(t)> 0 and v

(t) = 0.

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Let I = (0, t) denote the maximal interval of continuation of every semiclassical solution of (21)(22), under the restrictions

v > 0 and < v < 0 in I. (24)

From the definition ofI, it is clear that the solutions of (21) we consider have theproperty that v(t)< 0. Letting(t) =|v(t)|, problem (21)(22) can be rewrittenas

[q(t)(t)p1] = q(t)f(v), t I,

v(0) = (0, ), v(0) = 0.(25)

In analogy to [1] and [12] we give the following lemmas.

Lemma 2. Letv1 be a solution of (25)defined in its maximal intervalI1,, deter-mined by (24). For all t0 I1, and >0, there exists >0 such that, ifv2 is asolution of (25), with |v1(0) v2(0)|< , thenv2 is defined in [0, t0] and

sup[0,t0]

{|v1(t) v2(t)| + |v1(t) v2(t)|}< .

Proof. By using (q1)(q3), the proof of Lemma 2.3 of [12] for the special caseq(t) = tN1 can be repeated since it was used only the fact that f Liploc(0, )together with (f1). For a complete proof we refer to Lemma 4.2 of [3].

A natural energy function associated to solutionsv of (7) is given by

E(t) = p(t)

p + F(v(t)), = |v|, (26)

which is of class C1(I {0}), with E(0) = 0 and in I

E(t) =q(t)q(t)

p(t), E(t) E(s0) = t

s0

q(s)q(s)

p(s)ds, (27)

see Lemma 5.3 and Section 5 of [17] for more detailed properties.

Lemma 3. Suppose that fsatisfies alternatively either (f2) or (f3). Let v be asolution of (25). Then the following results hold.

(i) The limit := lim

ttv(t) (28)

belongs to [0, ) if(f2) holds; while = 0 if(f3) holds.

(ii) If > 0 andt , then limtt

v(t) = 0.

(iii) Ift = , then limt

v(t) = 0.

(iv) Let > d. If > , then there exists a unique value t, I such that

v(t,) = .

Proof. (i) Clearly the limit in (28) exists and is nonnegative, since v is strictlydecreasing and positive inIby (24). Suppose first that (f2) holds. Then [0, )by (22). Assume by contradiction that [, ). Then < v(t)< in I,and in turn, by (25) and (f2), from [q(t)|v(t)|p1] =q(t)f(v)> 0, it follows thatq|v|p1 is strictly increasing in I.

Distinguish now two cases: t < and t = . If t < , since v(t) = >0, then v

(t) = 0 by (24), and so q(t)|v(t)|p1 0 as t t . On theother hand (q|v|p1)(0) = 0, and this contradicts the fact that q|v|p1 is strictlyincreasing in I.

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While if t = , then I = R+ and by Lemma 5.3 of [17] the energy functionalongv defined in (26) is of class C1(R+), with

E(t) = q(t)q(t)

p(t)< 0 in I. (29)

ThereforeEadmits finite limit as t . By (26) also(t) has limit as t ,and

v(t) t

0, (30)

since [, ) by contradiction. Rewrite the equation in (25) in the equivalentform

[p1(t)]+q(t)

q(t)p1(t) = f(v(t)), tI. (31)

By (30) and (q2)q

q[(t)]p1

t0,

and in turn by (31)limt

[p1(t)] = f()> 0,

since [, ) by contradiction. This is impossible, since p1 > 0 in R+ and

p1(t)0 as t .Suppose now that (f3) holds and that >0 by contradiction. We can repeat

the above proof, with [, ) replaced by (0, ), and obtain the desired contradiction.(ii) If > 0 and t 0 and t =, arguing as in (i), case t =, and using (29), we obtain thevalidity of (30).

(iii) Ift = , by (i) and the fact thatEadmits limit as t thenv(t) 0

ast .

(iv) In this case the proof is an immediate consequence of the fact that v isstrictly decreasing in I by (24).

As in [1] define

I := {(d, ) : t < , = 0, v(t)< 0}. (32)

Lemma 4. Suppose thatfverifies either (f2) or (f3). Let v be a semiclassicalsolution of (25), inI defined in (24). If /I

, then for alltIq(t)

q(t) 0) (v(t) = 0). In the third

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case there is nothing to prove, that is Mis achieved in [t, t); ift = , then (30)holds by Lemma 3 (iii), that is M is again achieved in [t, t). Finally if > 0,

thenv (t) = 0 by Lemma 3, and in turn Mis again achieved in [t, t).Moreover

E(t) = p(t)

p + F() = F() 0. (34)

Indeed if (f2) holds, then [0, ) and E(t) =F() 0; while if (f3) holds,E(t) =F() = 0. Using (q2) and (24) t

t

q(s)q(s)

p(s)dsq(t)

q(t)Mp1

tt

[(v(s))]dsq(t)

q(t)Mp1V. (35)

Hence by (27) and (34)

F(V) = F(v(t))< E(t) = E(t) + tt

q(s)q(s)

p

(s)ds

q(t)q(t)M

p

1

V. (36)

Now by (34) and (35)

Mp

p =

p(T1)

p =E(T1) F(v(T1)) E(t) +

tT1

q(s)q(s)

p(s)ds + F

F+q(T1)

q(T1)Mp1V F+

q(t)q(t)

Mp1V.

(37)

By the assumption of contradiction (33)

q(t)

q(t)

V

F(V)

p[F+ F(V)]

p[F+ F(V)]1/p,

and so

p[F+ F(V)]

p V

F(V)[F+ F(V)]

q(t)q(t)

p

=p V

F(V)[F+ F(V)]

q(t)q(t)

p

V

F(V)[F+ F(V)]

q(t)q(t)

p1,

that is

F(V)

V

q(t)

q(t)

p

V

F(V)[F+ F(V)]

q(t)q(t)

p1.

Therefore by (36) we get

Mp1 >F(V)

V

q(t)

q(t)

p

V

F(V)[F+ F(V)]

q(t)q(t)

p1,

in other words

M

p >

V

F(V)[F+ F(V)]

q(t)q(t)

. (38)

Consequently by (37)

F Mp

p

q(t)q(t)

Mp1V =Mp1

M

p

q(t)q(t)

V

,

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and in turn by (36) and (38)

F > Mp1

VF(V)

(F+ F(V)) q(t)q(t)

Vq(t)q(t)

= Mp1Vq(t)

q(t)

F+ F(V)

F(V) 1

F(V)

V

q(t)

q(t) V

q(t)q(t)

F+ F(V) F(V)

F(V)

= F .

This is the desired contradiction.

Let

(v) := pN F(v) (N p)vf(v), v R+, (39)

be the function given in assumptions (1) and () of the Introduction, and alonga solution v of (25) in I letPbe defined as

P(t) := (N p){q(t)v(t)v(t)|v(t)|p2 +pQ(t)E(t)}, Q(t) = t

0

q(s)ds. (40)

ClearlyP(0) = 0 by (f1), since v (0) = 0 and Q(0) = 0.We present an inequality proved with the same technique of Lemma 2.4 of [12].

Lemma 5. (NiPucciSerrin) Assume1< p < N. Letv be a solution of (25) inI, given in (24). Then

P(t)

t0

q()[(v()) p2F(v())]d. (41)

Proof. By Lemma 5.3 of [17] the energy function E is of class C1(I) and by theregularity ofv in I we can differentiate Pin (40), obtaining in I by (29)

P(t) = (N p)

q|v|p + v

qv|v|p2

+pqEpQq

q |v|p

= (N p)q

pF(v) vf(v) +p

1

Qq

q2

|v|p

,

where we have used (25) and (26) for the last step. Adding and subtracting theterm (pNp)F(v), wherep

N =pN/(Np) represents the Sobolev critical exponent

for (25), as discussed and proved in Section 4 of [17], we get

P(t) = (N p)q

pN

N pF(v) vf(v) (pN p)F(v) +p|v

|p

1 Qq

q2

.

By (39) and the fact that Qq/q2 1 in R+ by (q1)(q3), we have

P(t) q(v) (pN p)(N p)qF(v) = q[(v) p2F(v)].

Finally (41) follows at once sinceP(0) = 0, as noted above.

We recall that throughout the section we continue to assume the validity of (f1)and (f4).

Theorem 1. Let1< p < N and= . Suppose thatf satisfies(f3), ()and

(f5) lim inf v0+

vp

F(v) = 0.

ThenI=.

13/33


Proof. Suppose by contradiction that I = . Hence

/I for all R

+

. (42)Takeand as required in (), and >0 so that > > d = 0 by (f3). Withoutloss of generality we suppose that is so close to 1 and >0 so close to 0 that

>

, 0 0 sufficiently smallby (f5) and the fact that (0) := limt0+(t) = 0 by (q3) since N > 1. Clearlyt < t< t, since v(t) = > = v(t), by (43) and the fact that v

< 0 in I,and

v(0) v(t)

t1,

since

0 =v(0) = limt0+

v(t) v(0)

t = lim

1v(t) v(0)

t.

Indeed if 1, that is if , then t 0.Integrating the equation in (25) on [0, t], with t (0, t), we obtain

q(t) [(t)]p1 = t

0

q()f(v())d, (44)

since(0) =|v(0)|= 0. Hence, putting

f(2) := max[,]

f(u), 2 [, 1],

we have f(2)> 0, since > >0. Therefore by (44)

q(t) [(t)]p1 max[0,t]

f(v(t)) t

0

q() df(2)Q(t),

and in turn

|v(t)|

f(2) Q(t)

q(t)

1/(p1)[f(2) t]

1/(p1),

sinceQ(t)/q(t) t by (q1). Integrating this inequality on [0, t], t0

v(s)ds t

0

[s f(2)]1/(p1)ds

by (24). Thus

v(t) + v(0) =(1 ) [f(2)]1/(p

1)

tp

p ,in other words

[f(2)]1/(p1) (1 )p tp

. (45)

We choose so close to 1 that (1 )f(2) 1. Therefore

[(1 )f(2)]1/(p1)

1

[(1 )f(2)]1/(p1)

1

(1 )

1/(p1) tp(1 )p

,

by (45). Consequently

t (1 )1+1/p(p)1/p

p+1f(2)

1/p=c

p+1f(2)

1/p, (46)

14/33


wherec is the constant given in ().Since /I andF= 0 by (f3), from Lemma 4

q(t)

q(t) > by (43) and (), there is 1 [, 1] such that

(1) := minv(v) 0.

By construction we now have

(v(t))

(1), if 0 < t < t,0, if t t t,||, if t > t.

(49)

Since 1< p < N, by (40), (24), Lemma 5, (49) and (q1) for all t t

p(N p)Q(t)E(t) P(t)

t0

+

tt

+

tt

q(s)(v(s))ds p2

t0

q(s)F(v(s))ds

(1)Q(t) ||Q(t) p2

F()Q(t),sinceF is strictly increasing in R+0 by (f3) and the assumption = , so that

F(v(s)) F() for all t (t, t). (50)

Hence by (46) for all t (t, t)

p(N p)E(t) (1)Q(c[

p+1f(2)]1/p)

Q(t) || p

2F(). (51)

We now treat the cases t < and t = separately.Assume first thatt < . For each >0 define

T := min{t+ , t},

so that

T (t, t]. (52)

By () we can take > / so large that by (51)

E(t) F() + 1

p

pin (t, T]. (53)

In particular fort = T

E(T)F() + 1

p

p> F(). (54)

We claim thatT= t+ < t. (55)

15/33


Assume for contradiction that T = t < , then v(T) = v(t) = 0. Indeed

= 0 in (28) by (f3) and Lemma 3 (i). Furthermore, since /I, thenv(t) = 0.

Hence by (26) and (50)E(T) =F(v(T)) F(),

which contradicts (54) and proves the claim (55).By (53) for all t (t, T] I

|v(t)|p

p + F(v(t)) F() +

1

p

p,

andv(t) =|v(t)|> / by (50). Integrating in (t, T] by (55)

v(t) v(T)>

(T t) = ,

that is v(T)< 0. This is impossible since v >0 in [0, t) and completes the proofin the caset < .

Assume next that t =. By (51), (26), (50) and the assumption 1 < p < N,for all t t

(N p)(p 1)|v|p (1)Q(t)

Q(t) || p

2F() p(N p)F(). (56)

Since (t, t+ 1) I, by (56) for all t(t, t+ 1)

(N p)(p 1)|v|p (1) Q(t)

Q(c+ 1) || pN F(),

where c is the number defined in (48), which depends only on , p, F, but isindependent of, and we have used (q1) to have that

Q(t) Q(t) Q(t+ 1) < Q(c+ 1) in (t, t+ 1),

by (42) and by Lemma 4. By () we can take so large that|v| in [t, t +1].By (24) and integration on [t, t+ 1], we get

v(t+ 1) =v(t) +

t+1t

v(s)ds= v(t) t+1t

|v(s)|ds = 0,

which contradicts the fact that v > 0 in I = R+ and completes the proof also in

the caset = .

Remark. Iff(v) vm as v 0+, with m > 1 by (f1), then (f5) holds if andonly ifm < p 1.

The assertion of Theorem 1 continues to hold also when

16/33


uniformly in any bounded interval ofR+0 . Furthermore taking sufficiently closeto so that > , where := (+ d)/2, by (iv) of Lemma 3 there is a unique

valuet I such thatv(t, ) = .

We claim that the function defined in (d, ) by

t

is not bounded as .Indeed, if there is a constant t, such that 0< t t 1

F() {p[F+ F()]}1/p

,

by (q4). Lemma 4 guarantees that I, concluding the proof when k p and k/p + /p 1 n,

R+, if = k p and k/p + /p > 1 n.

In particular in all the cases of (4) in which the parameters verify either

= k p and

k

p +

p = 1 n, (57)or

> k p and k

p+

p 1 n, (58)

condition (q4) is satisfied.For instance (57) holds when n = p = = = 3, s = 2 and k = 0 so that

q(t) = tanh2 t; while condition (58) is valid when n= p = 2, =s = = 1 andk= 0 with q(t) = (t2 + 4t)/2(t + 2).

Assumption (q4) in terms of the radial weights a and b of the original radialequation (2), becomes

17/33


(A5) limr

(r) = 0,

since Q(t(r)) =r

0 b(s)ds by (40), (13) and (12). In particular in the interestingsubcase of (1) given when g 1, assumption (A5) reduces to

(A5) limr

h(r) = 0,

where

h(r) =

n 1

r +

1

ph

h

see also [17].

We now establish the results contained in Theorems 1 and 2 when () is replacedby condition (1) of Section 1, which was introduced in [4] for the Laplacian equa-tion in a ball. Condition (1) is the analogue subcritical assumption used in [8] and

[22] for thepLaplacian equations with no weights. The results of [8] were extendedin [12] to Aequations, while those of [8] and [22] were extended to Aequationsin [1], with a unified proof and also with the introduction of the new subcriticalcondition ().

Theorem 3. Let = . Suppose that f verifies (f1) and (f3)(f5). If (1)holds, thenI =.

Proof. We proceed as in the proof of Theorem 3 until (45). Now

t [(1 )p]1/p

p1

f(2)

1/p=C

p1

f(2)

1/p,

which replaces (46) in the proof of Theorem 3. Proceeding as in the proof of

Theorem 3 until (51), which becomes

p(N p)E(t)(1)

Q(t) Q

C

p1

f(2)

1/p || p

2F().

Also in this case we distinguish the two cases t < and t = , and get thesame conclusions as before.

Theorem 4. Let(f1), (f4), (1) and(q4) hold. Iff verifies either(f2) or (f3),thenI=.

Proof. In analogy of the proof of Theorem 3, following the proof of Theorem 2, wearrive to the desired conclusion.

4. Relation between() and (1)when= . In this section we compare thetwo growth conditions () and (1), as done in [1] when the weight qis the standardweight rn1. In particular we shall show that in cases interesting in applications,condition () holds while (1) does not.

Proposition 1. Let= . If

liminfv

f(v) =k0 (0, ], (59)

then (1) implies().


18/33


Proof. Fix (0, 1) and first note that for all 2 [, 1] and v > d/

c[vp+1f(2v)]1/p =C

vp

1

f(2v)

1/p [(1 )vf(2v)]2/p (60)

By (59) we have lim infv vf(v) limv v liminfv f(v) = , so that

limv

vf(v) = .

Hence there isv0 / sufficiently large, where > dis the number given in (1),such that for all v v0

f(v)> 0 and vf(2v) 2vf(2v) 1

1 . (61)

Thus forv v0 by (60) and (61)

c[vp+1f(2v)]

1/p C vp1

f(2v)

1/p

,

that is

Q

c[vp+1f(2v)]

1/p

Q

C

vp1

f(2v)

1/p,

sinceQ is increasing by (q1). In turn by (1) we have

(1v)Q

c[vp+1f(2v)]

1/p

(1v)Q

C

vp1

f(2v)

1/p,

and the conclusion follows.

The two growth conditions (1) and () are independent, since also the reverseimplication of Proposition 1 holds, as shown in the next

Proposition 2. Let= . If

limsupv

vf(v) = k1 [0, ), (62)

then () implies(1).

Proof. As in the proof of Proposition 1, fix (0, 1). By (60) for all 2 [, 1]andv > d/ we have

C

vp1

f(2v)

1/p=

c[vp+1f(2v)]1/p

[(1 )vf(2v)]2/p. (63)

By (62) there is >0 such thatk1 + /(1 ), and in turn by (62) again thereis v0 / sufficiently large, where now > d is the number given in (), such

that for all 2 [, 1] and v v0

vf(2v)k1+

2

1

1 .

Hence by (63)

C

vp1

f(2v)

1/pc[v

p+1f(2v)]1/p.

Therefore, since now for all v v0 and 1 [, 1]

(1v)Q

C

vp1

f(2v)

1/p (1v)Q

c[v

p+1f(2v)]1/p

,

19/33


the conclusion follows at once, as above.

Remark. When assumption (62) holds with the limsup replaced by the limit, thenlimv f(v) = 0, and the existence problem could be solved with much simplertechniques. Moreover, the nonlinearities which frequently appear in applicationstend to infinity at infinity, a subcase of (59).

Hence in cases interesting in applications () is more general than (1). Inparticular under (59), Theorems 3 and 4 are immediate corollaries of Theorems 1and 2, since in general (59) together with (), (f1), (f4) and either (f2) or (f3)do not imply the validity of (62). This will be clarified in the next section.

5. Canonical nonlinearities in the case =. Since in conditions () and(1) only the behavior offat infinity is important, in the examples we present inthis section, we shall define the various nonlinearities only for large values of v.

We also recall that in the sequel 1 < p < N and =. To simplify the notationin () and (1) we shall denote by and 1 the main involved functions, namely

(v) := (1v)Q(c[vp+1f(2v)]

1/p), 1(v) := (1v)Q

C

vp1

f(2v)

1/p.

First consider

f(v) = vp

N1 +1

v for v v0> 0.

Let fbe defined in [0, v0] so that f L1[0, v0] C(R

+) Liploc(R+), and also in

such a way that fsatisfies (f1), (f4) and either (f2) or (f3) in its entire domainR+, and finally so that is locally bounded near v = 0. Clearly also (59) holdswithk0= , since 1< p < N. Moreover for all v v0

F(v) = F(v0) vp

N0

pN log v0+ v

p

N

pN+ log v.

Hence, settingc0:=

F(v0) log v0 vpN0 /p

N

Np N+ p, we have for all v v0

(v) = N pF(v) (N p)vf(v) = c0+ pNlog v.

Of course is positive for all vsufficiently large, say forv , with >max{d, v0}.Fix (0, 1). Then for all 1, 2 [, 1] and v /

(v) = [c0+pNlog(1v)] Q

c

pN12 v

p+pN +vp

2

1/p

asv , since Q(t) as t by (q1), and in turn () holds.

While we claim that (1) does not hold. Indeed,

limv

C

vp1

f(2v)

1/p= lim

vC

(1pN)/p2 v

1pN/p = 0, (64)

since 1< p < pN. Hence for all v / sufficiently large and for all 2 [, 1] we

have thatC

vp1/f(2v)1/p

(0, 1), and so also for all 1 [, 1]

1(v) q(1)C [c0+ pNlog(1v)] v(p1)/p

(2v)pN1 +

1

2v

1/p ,

20/33


by (q1). Therefore

0 limv1(v) q(1)CpN (1pN)/p

2 limv v1p

N

/p

log(1v) = 0,

and the claim is proved.

Remark. Since the number N, given in (q3), is strictly greater than 1, we claimthat for all (0, N 1) there is t0=t0()> 0 and two constants C1,, C2, > 0,depending on and N, such that

C1,tN+ Q(t) C2,t

N for t (0, t0). (65)

Indeed, fixed (0, N 1), by (q3) there is t0 = t0() > 0 such that for allt(0, t0)

0< N 1

t 0.

Let (1, pN),

f(v) =vpN1 + v1 for all v v0> 0,

and let again f be defined in [0, v0] so that f L1[0, v0] C(R

+) Liploc(R+),

and also in such a way that f satisfies (f1), (f4) and either (f2) or (f3) in itsentire domain R+, and finally so that is locally bounded near v = 0. Also (59) issatisfied with k0= . By (f1) for all v v0

F(v) = F(v0) +vp

N

pN+

v

vpN0

pN

v0

.

Hence (v) =c0+ c1v for all v v0, where

c0:= N p

F(v0)

vpN0

pN

v0

and c1:= p N+

N p

>0,

since < pN. Thus there is > max{d, v0} sufficiently large such that (v) 0for all v . Fix (0, 1) and put v0 = /. Therefore for all v / and 1,2 [, 1] we have

(1v) Q(c[pN12 v

p+pN + 12 vp+]1/p) ,

asv , namely () holds.Now, since < pN and p < p

N,

limv

C

vp1

f(2v)

1/p=C

(1pN)/p2 limv

v1p

N/p = 0.

21/33


Furthermore, since < pN, fix (0, N 1) so small that

< pN p

N p .

Since 1< p < N, by (65)2 there is v1= v1() / sufficiently large such that forallv v1 and 1, 2 [, 1] we have

(1v)Q

Cv

(p1)/p

[(2v)p

N1 + (2v)1]1/p

C2,C

N [c0+ (1v)c1] v

(p1)(N)/p

[(2v)p

N1 + (2v)1](N)/p

.

Thus the right hand side approaches zero as v since

+(p 1)(N )

p

(pN 1)(N )p

0, (66)

and let f be defined in [0, v0] so that f L1[0, v0] C(R+) Liploc(R+), and

also in such a way that f satisfies (f1), (f4) and either (f2) or (f3) in its entiredomain R+; and finally so that is locally bounded near v = 0. Here we assumethat c0 := F(v0) v

m0 /m > 0. Clearly f verifies (62) with k1 = 0. As before,

F(v) =c0+ vm/mfor all v v0, and so

(v) = c1+ c2vm, c1:= N pc0> 0, c2:= p N+ Np/m 0 by construction, namely (1) is valid. While as v

(v) c1 Q

c(m1)/p2 v

(p+m)/p

c1Q

c/

(p+1)/p2

, if m= p,

0, if m < p,

that is () does not hold.

Let

f(v) = ev, v v0,

with v0 = 0 in case (f3). Otherwise we takev0 > 0 and define f in [0, v0] so thatfL1[0, v0]C(R

+)Liploc(R+),F(v0)> 0, and also in such a way thatfsatisfies

(f1), (f2) and (f4) in its entire domain R+, and is locally bounded near v = 0.With the usual notationF(v) = c0 e

v for allv v0, withc0= F(v0) + ev0 andeitherF(v0) = 0 in case (f3) or F(v0)> 0 by construction in case (f2). Hence inboth casesc0> 0. For all v v0

(v) =c1 [N p + (N p)v]ev, c1= N pc0> 0.

22/33


Therefore there is > max{d, v0} so large that (v) 0 for all v . Fix

(0, 1). Moreover for all 2 [, 1] we have Cv1/pe(2v)/p as v , so

that also for all 1 [, 1]

1(v) = {c1 [Np + (N p)1v]e1v} Q

Cv

1/pe2v/p

asv , that is (1) is valid. While c[vp+1e2v]1/p 0 asv and so also

asv

(v) = {c1 [N p + (N p)1v]e1v} Q

c[v

p+1e2v]1/p

0,

sinceQ(0) = 0, that is () is not valid.

Finally we present two examples to which all Theorems 14 can be applied, sinceboth growth conditions () and (1) hold.

Letp < m < pN f(v) =vm1, v v0> 0, (67)

and letfbe defined in [0, v0] so thatfL1[0, v0] C(R

+) Liploc(R+), and also in

such a way thatfsatisfies (f1), (f4) and either (f2) or (f3) in its entire domain R+, is locally bounded near v = 0, and with the further property thatF(v0)> v

m0 /m

ifm = 0. Now set

c0(m) = F(v0)

vm0 /m, if p < m < p

N, m= 0,

log v0, if m= 0,

consequently forv v0

F(v) = c0(m) + vm/m, if p < m < pN, m= 0,

log v, if m= 0,

and so, putting c2:= p N+ Np/m, we have for all v v0

(v; m) = N pc0(m) +

c2v

m, if p < m < pN, m= 0,

p N+ N p log v, if m= 0.

Therefore there is >max{d, v0} so large that (v; m) 0 for all v . Indeedc2 > 0 when 0 < m 0 for p < m < 0; and of course

(v; 0) as v when m = 0.Let (0, 1). For all 1, 2 [, 1] and v /

(v) = (1v; m) Q

c(m1)/p2 v

(p+m)/p

as v .

Similarly forp < m p

1(v) = (1v; m) Q

C(1m)/p2 v

(pm)/p

as v .

While ifp < m < pNthere is >0 sufficiently small that

m < pN(m p)

N p . (68)

Now, for all 1, 2 [, 1] we have C(1m)/p2 v

(pm)/p 0 as v , and so by(65)1 for all v v1 /, with v1 large enough,

1(v) C1,m1 c2

C

(1m)/p2

N+ vm+(pm)(N+)/p

23/33


as v since Npc0(m)> 0 by construction, c2 >0 by the fact that m > p >1and

m +(p m)(N+ )p

>0

by (68). In conclusion also (1) is valid.

Assume that (67) holds with m > p and let f be defined in [0, v0] so thatf L1[0, v0]C(R

+)Liploc(R+), and also in such a way that f satisfies (f1),

(f4) and either (f2) or (f3) in its entire domain R+, that is locally bounded nearv= 0, and with the further property that F(v0)> v

m0 /mifm = 0. Then repeating

the same argument above, () and (1) hold if and only if

m < pN.

Indeed the sufficient part is proved above. For the necessary part ifm 0 then thepositivity of forces that m < pN, while if m < 0 obviously there is nothing to

prove. In this case all Theorems 14 can be applied.Hence () and (1) hold for (67) with m > p if and only ifm < pN, that is

they are subcritical growth conditions for fatin the sense of Sobolev embeddingwith weights.

Let 0 m < pN,

f(v) = vm1 log v, v v0> 0, (69)

and let f be defined in [0, v0] so that fL1[0, v0] C(R

+) Liploc(R+), and also

in such a way thatfsatisfies (f1), (f4) and either (f2) or (f3) in its entire domainR+, and is locally bounded near v = 0. As before, put

c0(m) = F(v0) + vm0 [1 m log v0]/m

2, if 0< m < pN,

log2

v0/2, if m= 0,and so for all v v0

F(v) = c0(m) +

vm[m log v 1]/m2, if 0< m < pN,

log2 v/2, if m= 0,

(v; m) = N pc0(m)+

vm[m(N p)(pN m)log v N p]/m

2, if 0< m < pN,

log v[N p log v 2(N p)]/2, if m= 0.

Hence there is >max{d, v0} so large that (v) 0 for all v , since m < pN.

Let (0, 1). For all 1, 2 [, 1] and v / we have

(v) = (1v; m) Qc[

m12 v

m+p log(2v)]1/p

as v .

Namely () holds.Now to prove the validity of (1), we distinguish two cases again for all 1,

2 [, 1] and v /. Ifm[0, p), then

1(v) = (1v; m)Q

C

1m2 v

pm 1

log(2v)

1/p as v .

While in the remaining case m [p, pN), we can argue as for the example (67),since

C

1m2 v

pm

log(2v)

1/p0

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asv . Therefore, taking >0 so small that

m +(p m)(N+ )

p >0,

by (65)1 forv sufficiently large we get

1(v) C1,(1v; m)[C(1m)/p2 ]

N+

vpm

log(2v)

(N+)/pC1,[C

(1m)/p2 ]

N+(p N)(1 pN/m)m1 v

m+(pm)(N+)/p

log(1v)

[log(2v)](N+)/p

,

and (1) follows at once letting v .

6. Preliminary Lemmas for the existence of radial ground states. Through-

out the section we assume that the nonlinearityfin (25) verifies assumptions (f1),(f2) and (f4). Let

I+ :={ : > 0}. (70)

Of course I+ and I are disjoint, where I is given in (32). We shall prove belowsome properties useful for the proof of the main existence Theorems 7 and 10.

Lemma 6. belongs to I+.

Proof. Letv be a solution of

[q(t)|v(t)|p1] = q(t)f(v), v(0) =, v(0) = 0,

defined in I = (0, t), given in (24). From (26), we get

E(0) =

p(0)

p + F(v(0)) =F() = 0, (71)thanks to (f2). Moreover,E is strictly decreasing in I by (27), hence E(t)< 0 inI by (71). Now, fix t0 I , then

F(v(t)) E(t)< E(t0)< 0 in (t0, t).

Hence, by letting t t ,we get F() E(t0)< 0, where is defined in (28). Inturn >0 by (f2), since F(0) = 0 by (f1). Thus I+.

Lemma 7. I+ is open in [, ).

Proof. Fix I+. Letv be the solution of (25), with replaced by , definedin its maximal interval I= (0, t), in the sense of (24). Clearly

(0, ) and v

(t) 0 as tt

(72)

by Lemma 3 (i), (ii) and the fact that I+. Thus

limtt

E(t) =F()< 0,

by (26) and (f2), since (0, ).First fix t0 I such that E(t0) < 0. If > 0 is chosen sufficiently close to

and v is the corresponding solution of (25), then by Lemma 2 we have that(0, t0] I, where I is the maximal interval of continuation of v. FurthermoreE(t0) E(t0)/2< 0. As in the proof of Lemma 6, this implies that is inI+.

Lemma 8. I is open.

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Proof. Fix I and let (k)kN be any sequence of positive numbers convergentto . Let v be the solution of (25), with v(0) = , in its maximal domain of

continuation I = (0, t); and similarly denote by vk the solution of (25), withvk(0) = k, in the maximal domain Ik = (0, tk), in the sense of (24). DenotebyE and Ek the energy functions along the solutions v and vk, respectively. Putc := E(t)/2. Clearly c > 0 by (26), since v

(t) < 0 and = 0 by the factthat I. By Lemma 2 of courseEk E 0 as k , so that there iss0 (t/2, t) such that 2c < E(s0) < 3c by (27), and for all k N sufficientlylarge

tk > s0, c Ek(s0) 4c, vk(s0) 2v(s0) . (73)

By (27), integrating on [s0, tk), we have by (q2)

|Ek(tk) Ek(s0)| q(s0)

q(s0) sup

[s0,tk)

p1k (s) tks0

k(s)ds,

wherek := |vk|. Now tk

s0

k(t)dt=

tks0

vk(t)dt= vk(s0)vk(tk)

dv vk(s0),

sincev k < 0 in Ik by (24). Hence for all k sufficiently large

|Ek(tk) Ek(s0)| q(s0)

q(s0)vk(s0) sup

[s0,tk)

p1k (t) M0 sup[s0,tk)

p1k (t), (74)

whereM0:= 2q(s0)v(s0)/q(s0). By (27) and (73)2 for all t [s0, tk)

pk(t) = p[Ek(t) F(vk(t))] p[Ek(s0) F(vk(t))] p

[4c + F],

whereF := maxv[0,]|F(v)|. In turn

sup[s0,tk)

p1k (t) [p(4c + F)]1/p

:= c,

and by (74) we obtain

|Ek(tk) Ek(s0)| cM0. (75)

Clearly (75) remains valid when s0 is replaced by any t in (s0, t) (t/2, t).Sincev(t) 0 ast t , being I

, then Ek(s0) c >0 by (73)2. Moreover,sinceF(vk(tk)) 0, being vk(tk)) , then

pk(tk)> 0 by (26), that is

vk(tk)< 0, tk 0.

Theorem 5. Assume (f1), (f3)(f5) and (), with = . Then (1) admits asemiclassical nonsingular radial crossing solutionu in the ballB, with

u(0) = >0, Du(0) =0, r < ,

u(x) = 0 and Du(x) x

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Proof. By virtue of Theorem 1 there is I and a corresponding solutionv of (7)in the interval I = (0, t), maximal in the sense of (24). Henceu(x) = v(t(|x|))

in B, r = r(t) by (12), is the required solution.

Remarks. Theorem 5 can be applied to all equations (4), including the classicalMatukuma equation, when f verifies (f5) and =.

In the recent paper [9] some existence, nonexistence and uniqueness results forradial ground states of some special cases of (2) are given when f >0 everywherein R+ but singular at u = 0. Moreover in [9] it is required that g 1, and h iscontinuous also at r = 0 and verifies an integral condition. In their prototype

f(u) = um + u, (77)

the main existence Theorem 1.1 of [9] can be applied provided that 1 0 if 1< p 2, or when(p 2)n+(p 1) + p < 0 ifp 2, that is in both cases when < 0, so that hmust be singular atr = 0. Hence Theorem 1.1 of [9] cannot be applied in this case,since in [9] the weight h is required to be continuous also at r = 0. In the famousMatukuma case, namely when g 1 and h(r) = (1 +r2)1, n = 3, p = 2, themain condition (1.4) of [9] again fails to hold, so that Theorem 1.1 of [9] cannot beapplied.

For (77) conditions (f1), (f3)(f5) and both () and (1) hold provided that

1< p < N, 1 p < m < 1, m < < pN 1.

Clearly Theorem 5 can be applied in both examples discussed above, and actuallyalso in the generalized Matukuma equations (see (1.4) of [17]), namely when g 1andh(r) = rp

/(1 + rp

)/p

, >0, and

n 2, p 2, with N=pn + p

p + p >1.

Hence in the classical subcase p = 2 and n= 3 it results N= 2(1 +)/ > 2. Inany case here 1 < p < Nif and only if 1 < p < n.

We now present another result when is possibly finite.

Theorem 6. Assume that (f1), (f3), (f4), (q4) and () hold. Then (1) admitsa semiclassical nonsingular radial crossing solutionu in the ballB, satisfying(76).

Proof. Here by virtue of Theorem 2 there is I, so that 0 < < , and acorresponding solution v of (25) in the interval I = (0, t), maximal in the sense

of (24). Henceu(x) = v(t(|x|)) inB,r = r(t) by (12), is the required solutionalso in this case.

Remarks. When fis of the required type, Theorem 6 can be applied to all equa-tions (4), with exponents verifying (5) and either (57) or (58), but not, for instance,to the Matukuma equation.

Some existence and nonexistence results for radial ground states of special casesof (2) are given in [5] in the case in which f is nonnegative foru >0 small. Theyprove existence of nontrivial positive radial solutions of (1) in the interesting casein which the continuous nonlinearity fmay depend onr, but is continuous also atu= 0. Furthermore, they consider (2) when g(r) = rk,k + n p >0, h(r) rh(r)

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asr with k+ 1 > 0 and f(r, u) h(r)um foru >0 sufficiently small and

r sufficiently large, with m > p 1, andhverifying some integral conditions.As a corollary of Theorem 6 we obtain existence of crossing solutions for the

pHessian operator Hp1, p Z, according to the notation in [23] and [5], see also[17]. For instance, in the first prototype studied in Theorem 5.2 of [5], that is

div(r2p|Du|p2Du) +|u|m1u

1 + r = 0, r >0, p >1,

u(0) = >0, u(0) = 0,(78)

existence of crossing solutions is proved by Theorem 6, provided that

0 1, Theorem 5.2(M1) of

[5] can also be applied so that problem (78) has no positive solutions. ThereforeTheorem 6 adds more information in the mentioned nonexistence result of [5]. Forthe same reason, also in the case >0, p 2 and m >1 Theorem 6 adds furtherinformation with respect to Theorem 5.2(M4) of [5] since

(p 1)(n + 2)

n 2(p 1) >

(n )p [n 2(p 1)]

n 2(p 1) and p 2(p 1).

In the other special case treated in [5], namely BattFaltenbacherHorst case, thatis when (4) reduces to

div(r2p|Du|p2Du) + rp

(1 + rp)/p|u|m1u= 0, r >0, p >1,

u(0) = >0, u(0) = 0,(80)

Theorem 6 shows that problem (80) admits crossing solutions when

p >2, >0, n >2(p 1), 1< m < (n + p)p n + 2(p 1)

n 2(p 1) ,

while in the somewhat complementary case

21, n >2(p 1), m (n + p)p n + 2(p 1)

n 2(p 1) ,

Theorem 5.3 of [5] guarantees that (80) has a ground state solution, so that nowthe picture is more detailed.

We now turn to the more delicate case in which the nonlinearities treated areof the type (f2).

Theorem 7. Assume that(f1), (f2), (f4), (q4) and()hold.Iff is continuous also atu = 0 andf(0) = 0, then (1) admits a semiclassical

nonsingular radial ground stateu, with < u(0) = < , which is compactlysupported inRn if

0+

du

|F(u)|1/p 0,

0+

du

|G(u)|1/p =, (82)

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whereG: [0, ) R, withG(0) = 0, is nondecreasing. When (81) holds, then thecompactly supported solutionu is also a semiclassical nonsingular radial solution

of the free boundary problem (3) for someR >0.When lim supu0+f(u) < 0, then problem (1) admits no semiclassical non

singular radial ground states, while (3) has a semiclassical nonsingular radialsolution for someR >0.

When f is singular at u = 0 and lim supu0+f(u) 0, then either (1) has apositive semiclassical nonsingular radial ground state or (3)has a radial solutionfor someR >0. If (82) is satisfied, then the first case occurs; if (81) is satisfied,the second case occurs.

Moreover, if for some 0, (83)

then the solution u is regular, that is Du is Holder continuous at x= 0, withDu(0) =0; while if [1, p), thenu is Holder continuous atx= 0. In both cases

u W1,ploc(Rn),

when also 1< p n.

Proof. As already noted, I+ and I are disjoint. By Theorem 2 and Lemmas 68,the sets I+ and I are open and not empty. Hence there is / I+ I, whosecorresponding solution v of (25) is positive in the maximal interval I, given by(24). Since / I+, then = 0 by (70). Since / I

, then either t = ort < . In both cases v

(t) = 0 by Lemma 3 (iii) and the fact that / I

.In the first case v is a positive semiclassical radial ground state of (25) and sou(x) =v(t(|x|)) is a semiclassical radial ground state of (1). In the second case

u(x) is a solution of (3) with R = R = r(t). In particular in this latter casewhenfis continuous also atu = 0, withf(0) = 0, the solution, when it is extendedto all x, with |x| > R, by the value 0, becomes a compactly supported radialground state of (1).

In conclusion we have shown that, iffis continuous also atu = 0, withf(0) = 0,problem (1) admits a semiclassical nonsingular radial ground state u, with 0 such thatf(u) 0 for0< u < .Of course when both (f2) and (f6) hold, then < .

Theorem 11. Assume (A1)(A4), (f1), (f2) and (f6), and that 1 0 is sufficiently small.Finally, if

(A6)

1p

a

a + 1

p b

b

r0 b(s)ds

b(r) N 1

N in R+,

then problem (1) does not admit any positive radial semi-classical non-singularground state.

Proof. Assume by contradiction that there exists a semiclassical nonsingular po-sitive radial ground state u of (1), and denote by v(t) =u(r(t)) the correspondingclassical ground state of (7). By (29) and (7) we get that

[Q(t)E(t) + q(t)|v(t)|p2v(t)v(t)] = qEQq

q |v|p + q|v|p qvf(v).


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Using (26), by integration of the above equality we obtain

Q(t)E(t) + q(t)|v(t)|p

2

v(t)v(t)

=

t0

q

F(v) vf(v) +

Qq

q2 +

1

p

|v|p

ds,

(87)

for allt >0 and R. Now take = 1/( + 1), where is given in (85). Then by(85) and the fact that (A6) is equivalent to (q5), we obtain in R+

Qq

q2 +

1

p

1

+ 1

1

pN0.

Hence by (86) the right side of (87) is strictly negative. In the left hand side of (87)we have that

limt

q(t)|v(t)|p2v(t)v(t) = 0

by Proposition 6.1 of [17], since v is a ground state of (7). On the other hand for tsufficiently large

[Q(t)E(t)] q(t)|v(t)|p

1

p

Qq

q2

q(t)

pN|v(t)|p 0. Furthermore by Corollary 5.8 of [17]a ground state of (7), with fgiven by (8), is positive in the entire Rn if and onlyifm p 1, and so the claim is proved.

Finally, the regularity property under (83) follows exactly as in the proof ofTheorem 7.

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Remarks. 1. As noted by Ni and Serrin in the Remark at the end of Theorem 3.3of [13], iff(u) =u(u), C1(R+), (u)< 0 for u small, then

( + 1)F(u) uf(u) =

u0

t+1(t)dt. (89)

Consequently, condition (86) holds if (u) 0 in R+, and (u) > 0 for u > 0sufficiently small.

If, for example, we consider f(u) = um + u, 1 < m < , we see that(u) = 1 um and

( + 1)F(u) uf(u) =m

m + 1um+1.

Thus (86) holds since m < .2. As used in the proof of Theorem 11, assumption (A6), in terms of the radial

weightsa and b of the original radial equation (2), is equivalent to condition (q5) in

terms of the new weighted equation (7), where q is given by (13), since Q(t(r)) =r0

b(s)dsas noted in the Remark before Theorem 3.In particular when g 1 assumption (A6) becomes

(A6)

n 1

r +

1

p

h

h

r0

sn1h(s)dsrn1h(r)

N 1

N .

Finally, we give some examples of functions g and h, for which (A1)(A6) or (q1)(q5) hold.

3. Of course in the case

g(r) 1, h(r) = r,

then as noted in Section 2 the main structure assumptions (A1)(A4) hold if (19)

is satisfied, with N >1 given in (20). In this case q(t) =tN

1

and so (q4) and (q5)trivially hold. In conclusion all (A1)(A6) are valid.Another interesting example is given by

g(r) 1, h(r) = log(1 + r),

where (A1)(A5) hold, with N =p(n+ 1)/(p+ 1) > 1. Furthermore, whenp= 2andn = 3, it is not hard to see that also (A6) is valid.

Moreover another example is given by

g(r) = h(r) = r1n(er 1), q(t) =e

t 1, r, t R+0.

Indeed (q1), (q2) and (q4) trivially hold, q(0) = 0, (q3) is satisfied with N = 3/2.Finally (q5) is verified whenn = 3 andp = 2, since Qq/q2 is an increasing functionsuch that limt

0+Qq

/q2 = 1/3(= (N 1)/N) and limt

Qq/q2 = 1.4. Actually Theorem 11 is the special case c = 1 1/Nof the following more

general nonexistence result:Assume(A1)(A4), (f1), (f2) and(f6), and that1< p < N. If

(A6)

1

p

a

a +

1

p

b

b

r0

b(s)ds

b(r) c inR+,

with

0< c 1

p

1

pN,

and if

pF(u) (pc 1)uf(u) in R+,

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with strict inequality when u > 0 is sufficiently small, then problem (1) does notadmit any positive radial semiclassical nonsingular ground state.

The proof is exactly the same as that of Theorem 11, where now c replaces theprevious main number 1 1/N. Note that here + 1 = p/(pc 1) pN, andc isany positive number, such that

1

N 1 c 0.

Hence (f1), (f2), (f4) and (f6) are also satisfied by (90), since 1< m < , with= .

The study of uniqueness of radial ground states of (1) or non singular solutionssemiclassical of (3) is very delicate. The first results are contained in [17], wherethe nonlinearityfis assumed to have a sublineargrowth at infinity, see also relatedresults, even if more specific, given in [9]. In particular in [17] it was proved forthe special nonlinearity (89) that the corresponding equation (1) admits at mostone semiclassical radial ground state u, with 0< u(0)< =, when (6) holds,namely when

p 2, 1< m 0 and 0, when f satisfies (6).

Acknowledgements. The authors were supported by MIUR projectMetodi Va-riazionali ed Equazioni Differenziali non Lineari.

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Received April 2005; revised November 2005.E-mail address: [email protected] address: [email protected] address: [email protected]

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