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EXISTENCE RESULTS FOR OPERATOR EQUATIONS

INVOLVING DUALITY MAPPINGS

VIA THE MOUNTAIN PASS THEOREM

JENICA CRINGANU

We derive existence results for operator equations having the form

Ju= Nfu,

by using the Mountain Pass Theorem. Here J is a duality mapping on a realreflexive and smooth Banach space, compactly imbedded in a Lq-space, and Nfis the Nemytskii operator generated by a Caratheodory functionf which satisfiessome appropriate growth conditions.

AMS 2000 Subject Classification: 35J20, 35J60.

Key words: Duality mapping, Mountain Pass Theorem, Nemytskii operator, p-Laplacian.

1. INTRODUCTION

In [9] the existence of the weak solution in W1,p

0 (), 1 < p < +, for

the Dirichlet problem

pu= f(x, u) in RN,(1.1)

u= 0 on (1.2)

was obtained using (among other methods) the Mountain Pass Theorem.

It is well known that the operator p : W1,p0 () W

1,p() is in

fact the duality mapping on W1,p0 () corresponding to the gauge function(t) = tp1. In this paper we generalize the results from [9] by consideringoperator equations of the form

(1.3) J

u= Nf

u,

where, instead of the special duality mapping (p), we consider the case ofan arbitrary duality mapping. More precisely, we consider equation (1.1) inthe functional framework below.

MATH. REPORTS 11(61), 1 (2009), 1120

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12 Jenica Crnganu 2

(H1) Xis a real reflexive and smooth Banach space, compactly imbedded in

Lq(), where 1< q 0 is constant and b Lq

().

By solution of equation (1.3) we mean an element u Xwhich satisfies

(1.5) Ju= (iNfi)u,

where i is the compact imbedding of X into Lq() and i : Lq

() X isits adjoint.

Notice that by (1.4) the operator Nf is well-defined, continuous and

bounded from Lq() into Lq

() (see, e.g., [8]), so that iNfi is also well-defined and compact from X into X.

Moreover, the functional : Lq() R, defined by(u) =

F(x, u)dx,

whereF(x, s) =s0 f(x, t) dt is C

1 on Lq(), hence on Xand (u) =Nfuforall u X.

Notice also thatF is a Caratheodory function and there exists a constantc1> 0 and a function c L

q(), c 0, such that

(1.6) |F(x, s)| c1|s|q +c(x), x , s R.

Let us remark that (1.5) is equivalent to

(1.7) Ju, v= Nf(iu), ivLq(),Lq ()=

f(x, u)vdx, v X.

By Asplunds theorem, Ju = (u) for each u X, where (u) =u0 (t)dt and : X P(X) is the subdifferential of in the sense of

convex analysis, i.e.,

(u) ={x X : (v) (u) x, v u for all v X}

(see, e.g., [4], [5] or [12]). Since is convex and X is smooth, is Gateauxdifferentiable on X and (u) = {(u)} for all u X. So, Ju = (u)

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3 Existence results for operator equations involving duality mappings 13

for all u X and, by continuity of J (see (H2)), we have C1(X,R).

Consequently, the functional F :X R defined by

F(u) = (u) (u) =

u0

(t)dt

F(x, u)dx, u X,

is C1 on X and F(u) = (u) (u) =Ju Nfu for all u X. Therefore,u Xis a solution of equation (1.3) if and only ifu is a critical point for F,i.e.,F(u) = 0.

To prove that Fhas at least a critical point in Xwe use the MountainPass Theorem, which we recall below.

Theorem 1.1. LetXbe a real Banach space and assumeI C1(X,R)satisfies the Palais-Smale(PS) condition. SupposeI(0) = 0 and that

(i) there are constants, >0 such thatI|u= ;(ii) there is an elemente X, e> such thatI(e) 0.

ThenIpossesses a critical valuec . Moreover,

c= infg

maxug([0,1])

I(u)

where ={g C([0, 1], X) :g(0) = 0, g(1) =e}. (It is obvious that each criti-cal pointu at levelc (I(u) = 0, I(u) =c) is a nontrivial one.)

For the proofsee, e.g., [2], [11] or [13].Let us recall that the functional I C1(X,R) is said to satisfy the

(PS) condition if any sequence (un) X for which (I(un)) is bounded and

I(un) 0 as n , possesses a convergent subsequence.

2. THE MAIN EXISTENCE RESULT

First, we establish some preliminary results.

Proposition 2.1. If (un) X is bounded andF(un) 0 as n ,

then(un) has a convergent subsequence.

Proof. This result is proved in [9], Lemma 2.

Theorem 2.1. Assume that there are >1 ands0 > 0 such that

(2.1) F(x, s) sf(x, s), x , |s| s0,

(2.2) lim inf t

t0(s)ds

t(t) >

1

.

ThenF satisfies the(PS) condition.

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Proof. According to Proposition 2.1 it is sufficient to show that any

sequence (un) Xfor which (F(un)) is bounded andF(un) 0, is bounded.As in the proof of Theorem 15 in [9], there exists a constant k > 0

such that

(2.3) (un) 1

f(x, un) undx k.

By F(un) 0, as in the proof of the same theorem, there exists n0 Nsuch that Jun, un

f(x, un)un

un, () n n0,which implies

(2.4) 1

(un)un +1

f(x, un)undx 1

un, () n 1.

From (2.3) and (2.4) we obtain

(un) 1

(un)un

1

un k, () n n0,

so that un0

(s)ds 1

(un)un

1

un k, () n n0

or, equivalently,

(2.5) (un)un un

0 (s)ds(un)un

1

1(un)

k.

By condition (2.2) there are constants > 1 and t0 > 0 such thatt0(s)ds

t(t) , t t0.

Ifun there exists n1 N such thatun t0 for n n1 and thenun0 (s)ds

un(un) , n n1.

From (2.5) we obtain

(un)un

1

1

(un)

k, n n1.

a contradiction (because > 1 and (un) ).Consequently, (un) is bounded in X.

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Theorem 2.2. Assume that there exists >1 such that

(2.6) limt

t0(s)ds

t = 0

and either

(i) there existss1 > 0 such that

(2.7) 0< F(x, s) sf(x, s), x , s s1,

or

(ii) there existss1 < 0 such that

(2.8) 0< F(x, s) sf(x, s), x , s s1.

ThenF is unbounded from below.

Proof. We are going to prove the sufficiency of condition (i) (similararguments may be used if (ii) holds).

Let u X, u > 0 (in fact i(u) > 0) be such that meas (M1(u)) > 0,where

M1(u) ={x :u(x) s1} (in facti(u)(x) s1).

We shall prove that F(u) as .For 1, from the proof of Theorem 16 in [9] we have

F(x,u)dx k1(u) k2,

wherek1(u)> 0, k2 > 0 are constants. Therefore,

F(u) = (u)

F(x,u)dx (u) k1(u) +k2

=

u0 (t)dt

k1(u)

+k2.

By condition (2.6) there are constants (0, k1(u)) and t0 > 0 such thatt0(s)ds

t u , t t0,

and then

F(u) u0 (t)dt

(u) u k1(u) +k2

( k1(u)) +k2

for u t0. Since k1(u)< 0, we have F(u) as .

Remark 2.1. By Theorem 2.2, sinceFis unbounded from below, for each >0 there exists e X withe> such that F(e) 0.

Now we are in a position to prove the main result of this section.

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Theorem2.3. Assume that hypotheses(H1), (H2)and(H3)hold. More-

over, assume that(i) there are >1 ands0>0 such that

(2.9) 0< F(x, s) sf(x, s), x , |s| s0;

(ii) lim inft

t0(s)ds

t(t) >

1

;

(iii) limt

t0(s)ds

t = 0;

(iv) there arer1, r2 (1, ), r1 < r2, andc0> 0 such that

F(u) c2ur1X c3u

r2X () u X withuX< c0,

wherec2, c3 > 0 are constants.Then equation (1.3) has at least a nontrivial solution inX.

Proof. Let us notice that condition (iv) is suggested by [10].It is sufficient to show that F has at least a nontrivial critical point

u X. To do it, we shall use Theorem 1.1. Clearly, F(0) = 0. By (i), (ii) andTheorem 2.1,Fsatisfies the (PS) condition. By (iv) we have

F(u) ur1X

c2 c3ur2r1X

for all u X, withuX< c0,hence there are constants , >0, < c0 such

that F(u) >0 provided that uX= is small enough.Finally, from (i), (iii) and Theorem 2.2 (see also Remark 2.1) there is an

element e X,e> , such that F(e) 0, and the proof is complete.

Remark 2.2. A sufficient condition in order to have (iv) is the existenceof real numbers r1, r2, 1< r1 < r2, such that

(iv1) lims0

s0 (t)dt

|s|r1>0,

(iv2) lims0

f(x, s)

|s|r21 0 is small enough while,

by (iv2),

F(x, u)dx c3ur2 ,

provided that u > 0 is small enough (here, c2, c3 > 0 are constants). Con-sequently,F(u) c2u

r1 c3ur2 for u> 0 sufficiently small.

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3. EXAMPLES

Example3.1 Consider the Dirichlet problem (1.1), (1.2), where 1 < p p,

+ ifNp.

Let us remark that thep-Laplacian operator p: W1,p0 () W

1,p()defined by

pu= Ni=1

xi

|u|p2

u

xi

, u W1,p0 (),

or, equivalently,

pu, v=

|u|p2u v, u, v W1,p0 (),

is the duality mapping

J: W1,p0 () W

1,p()

corresponding to the gauge function (t) =tp1 (see, e.g., [9] or [12]).

On the other hand, the Nemytskii operatorNfis continuous and boundedfrom Lq() into Lq

().By solution of the Dirichlet problem (1.1), (1.2) we mean an element

u W1,p0 () which satisfies

(3.2) pu=

iNfi

u

in W1,p

(), or, equivalently,

|u|p2uv=

f(x, u)v, v W1,p0 (),

where i is the compact imbedding of W1,p0 () into Lq() and i : Lq

()

W1,p

() is its adjoint. Consequently, (3.2) may be equivalently written as

(3.3) Ju= Nfu

(here, by Nfwe mean iNfi).

We shall formulate sufficient conditions for equation (3.3) to admit a non-

trivial solution, via Theorem 2.3. TakeX=W1,p0 () with RN, N2, a

bounded domain with smooth boundary, 1 < p

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t 0, f : R R a Caratheodory function which satisfies the conditions

below. The growth condition (3.1). There are > p and s0 > 0 such that

(3.4) 0< F(x, s) sf(x, s), x , |s| s0.

(3.5) lim sups0

f(x, s)

|s|p2s< 1 uniformly in x ,

where 1= inf

vp1,pvp0,p

:v W1,p0 (), v= 0

is the first eigenvalue ofp

in W1,p0 ().

SinceW1,p0 () is a reflexive and smooth Banach space, compactly imbed-ded in Lq(), hypothesis (H1) of Theorem 2.3 is satisfied. Since J =p :

W1,p0 () W

1,p() is continuous and satisfies condition (S+) (see, e.g., [9])hypothesis (H2) of Theorem 2.3 is satisfied, too. By the growth condition(3.1), hypothesis (H3) of Theorem 2.3 is also satisfied. Since > p, conditions(ii) and (iii) are satisfied. Finally, condition (3.5) implies condition (iv) inTheorem 2.3 (see, e.g., [9]).

Consequently, Theorem 2.3 applies and gives the already known result(see, e.g., [9]) on the existence of a nontrivial solution for problem (1.1), (1.2).

Example3.2. Consider the Neumann problempu+ |u|

p2u= f(x, u) in ,(3.6)

|u|p2u

n= 0 on ,(3.7)

where 1 < p < , RN, N 2, is a bounded domain with smoothboundary and f : R R is a Caratheodory function which satisfies thegrowth condition (3.1).

By solution of the Neumann problem (3.6), (3.7) we mean an elementu W1,p() which satisfies

(3.8) |u|

p2

uv+ |u|

p2

uv= f(x, u)v, v W

1,p

().

Assume X=W1,p() is endowed with the norm

|||u|||p1,p= up0,p+ |u|

p0,p, u W

1,p(),

which is equivalent to the standard norm on the space W1,p() (see [6]).

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In this case, the duality mappingJon W1,p(), ||| |||1,pcorrespondingto the gauge function (t) =tp1 is defined (see [6]) by

J:

W1,p(), ||| |||1,p

W1,p(), ||| |||1,p

,

Ju= pu+ |u|p2u, u W1,p().

It is easy to see that u W1,p() is a solution of problem (3.6), (3.7) in thesense of (3.8) if and only if

(3.9) Ju= Nfu

(here, byNfwe also meaniNfi, whereiis the compact imbedding of (W

1,p(),

||| |||1,p) into Lq() and i :Lq

()

W1,p(), ||| |||1,p

is its adjoint).We shall formulate sufficient conditions for equation (3.9) to admit a

nontrivial solution, via Theorem 2.3.Take X = W1,p() with RN, N 2, a bounded domain with

smooth boundary, 1< p p and s0 > 0 such that

(3.10) 0< F(x, s) sf(x, s), x , |s| s0.

(3.11) lim sups0

f(x, s)

|s|p2s< 1 uniformly in x ,

where1= inf

|||v|||p1,p

vp0,p:v W1,p(), v= 0

.

Since

W1,p(), ||| |||1,p

is a reflexive and smooth Banach space, com-pactly imbedded inLq() (see [6]), hypothesis (H1) of Theorem 2.3 is satisfied.Since J :

W1,p(), ||| |||1,p

W1,p(), ||| |||1,p

is continuous and satisfies

condition (S+) (see, e.g., [6]), hypothesis (H2) of Theorem 2.3 is satisfied, too.By the growth condition (3.1), hypothesis (H3) of Theorem 2.3 is also satisfied.Since > p, conditions (ii) and (iii) are satisfied. Finally, condition (3.11) im-plies condition (iv) in Theorem 2.3 (see, e.g., [6]). Consequently, Theorem 2.3applies and gives the already known result (see, e.g., [6]) on the existence of anontrivial solution for problem (3.6), (3.7).

REFERENCES

[1] R.A. Adams, Sobolev Spaces. Academic Press, New YorkSan FranciscoLondon, 1975.[2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory

and applications. J. Funct. Anal. 14 (1973), 349381.

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[3] H. Brezis,Analyse fonctionelle. Masson, Paris, 1983.

[4] F.E. Browder,Problemes non-lineaires. Les Presses de lUniversite de Montreal, 1964.[5] Ioana Cioranescu,Duality Mapping in Nonlinear Functional Analysis. Publishing Houseof the Romanian Academy, Bucharest, 1974. (Romanian)

[6] J. Crnganu, Variational and topological methods for Neumann problems with p-Lapla-cian. Comm. Appl. Nonlinear Anal. 11 (2004), 138.

[7] J. Crnganu and G. Dinca, Multiple solutions for a class of nonlinear equations involvinga duality mapping. Differential Integral Equations21 (2008), 265284.

[8] D.G. de Figueiredo, Lectures on Ekeland variational principle with applications anddetours. Tata Institut of Fundamental Research, Springer-Verlag, 1989.

[9] G. Dinca, P. Jebelean and J. Mahwin,Variational and Topological methods for Dirichletproblems withp-Laplacian. Portugaliae Math. 58 (2001), 339378.

[10] G. Dinca and P. Matei,Multiple solutions for operator equations involving duality map-pings on Orlicz-Sobolev spaces via the Mountain Pass Theorem. Rev. Roumaine Math.Pures Appl.53 (2008), 419438.

[11] O. Kavian, Introduction a la theorie des points critiques. Springer Verlag, 1993.[12] J.L. Lions, Quelques methodes de resolution des problemes aux limites non-lineaires.

Dunod-Gauthier, Villars, Paris, 1969.[13] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Springer,

1989.[14] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Dif-

ferential Equations. CBMS Reg. Ser. Math. 65. Amer. Math. Soc., Providence, RI,1986.

Received 14 November 2008 University of Galat i

Department of Mathematics

800008 Galati, Romania

[email protected]

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