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Hindawi Publishing CorporationBoundary Value Problems Volume 2009, Article ID 532546, 7 pages doi:10.1155/2009/532546 Research Article Infinitely Many Solutions for a Semilinear Elliptic

Hindawi Publishing Corporation

Boundary Value Problems

Volume 2009, Article ID 532546, 7 pages

doi:10.1155/2009/532546

Research Article

Infinitely Many Solutions for a Semilinear Elliptic Equation with Sign-Changing Potential

Chen Yu and Li Yongqing

School of Mathematics and Computer Sciences, Fujian Normal University, Fuzhou 350007, China

Correspondence should be addressed to Chen Yu,chenyusx@163.com

Received 23 March 2009; Accepted 10 June 2009

Recommended by Martin Schechter

We consider a similinear elliptic equation with sign-changing potential−Δu − V xu  fx, u,

u ∈ H1RN , where V x is a function possibly changing sign inRN Under certain assumptions

on f, we prove that the equation has infinitely many solutions.

Copyrightq 2009 C Yu and L Yongqing This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In this paper, the existence of solutions of the following elliptic equation:

− Δu − V xu  fx, u, u ∈ H1

RN

P

is studied, where V x is a function possibly changing sign, f is a continuous function on

RN× R

ProblemP arises in various branches of applied mathematics and has been studied

extensively in recent years For example, Rabinowitz 1 has studied the existence of a nontrivial solution of this kind of equation on a bounded domain Lien et al 2 studied the existence of positive solutions of problemP with V x ≡ λ λ is a positive constant and fx, u  |u| p−2u And Grossi et al. 3 established some existence results for −Δu 

λu  axgu, where ax is a function possibly changing sign, gu has superlinear growth and λ is a positive real parameter; he discussed both the cases of subcritical and critical growth for gu and proved the existence of linking type solutions.

Cerami et al 4 prove that the problem P has infinitely many solutions, where

a x is a regular function such that lim inf |x| → ∞ a x  a∞ > 0 and some suitable decay

assumptions, f x, u  |u| p−2u Kryszewski and Szulkin 5 considered the existence of

a nontrivial solution of P in a situation where fx, u and V x are periodic in the x-variable, f x, u is superlinear at u  0 and ±∞, and 0 lies in a spectral gap of −Δu  V

If in addition f x, u is odd in u, P has infinitely many solutions.

In6, Zeng and Li proved existence of m − n pairs of nontrivial solutions m > n, m and n are integers of P, under the assumption that V x is a function possibly changing

sign inRN and fx, u satisfies some growth conditions.

In this paper, we prove the existence of infinitely many solutions ofP, under the assumption that V x is a function possibly changing sign in R N and fx, u also satisfies

some growth conditions One difficulty in considering problem P is the loss of compactness because ofRN ; the other is that V x may change sign, which leads to difficulty in verifying

the Palais-Smale condition and applying the well-known theorem

Notation We use the following notations A strip region is a domain like this: for d >

0,  Ω  {x ∈ R N;−d < xi < d at least for some fixed i } V x  Vx − V−x, where

V± max{±V x, 0} Ω1 {x ∈ R N ; V−x / 0}, Ω2 {x ∈ R N ; V−x  0}.

X is defined as the completion of DRN with respect to the inner product

u, v1:



RN



∇u · ∇v  V−xuvdx. 1.1 The functional associated withP is

I u : 1

2



RN |∇u|2 V−xu2dx−1

2



RN

Vxu2dx−



RN

F x, udx, 1.2

for u ∈ X, where Fx, u u

0f x, tdt.

Our fundamental assumptions are as follows:

A1 Vx ∈ L N/2RN , meas{x ∈ R N ; Vx / 0} > 0 V−x ∈ L∞RN , Ω2 is a strip region, lim|x| → ∞ V−x  a > 0 in Ω1

A2 f ∈ CR N × R and there are constants C1 > 0 and 2 < p ≤ q < 2∗ such that

|fx, t| ≤ C1|t| p−1 |t| q−1

A3 There exists α > 2 such that 0 < αFx, t ≤ tfx, t for every x ∈ R N and t / 0

A4 lim|x| → ∞sup|t|≤r |fx, t|/|t|  0 for every r > 0.

A5 For any t ∈ R, fx, t  −fx, −t.

Here 2∗denotes the critical Sobolev exponent, that is, 2∗  2N/N − 2 for N ≥ 3 and

2∗ ∞ for N  1, 2.

Theorem 1.1 Under the assumptions A1–A5, P possesses infinitely many solutions on X.

Remark 1.2 It is easily seen that A2–A5 hold for nonlinearities of the form fx, t 

k

i1a ix|t| p i−2t with 2 < p i < 2∗ and for i  1, , k, the nonnegative function aix ∈

L∞RN , lim |x| → ∞ a ix  0.

Boundary Value Problems 3

2 Preliminaries

We define the Palais-Smaledenoted by PS sequences, PS-values, and PS-conditions

in X for I as follows.

Definition 2.1cf 7 i For c ∈ R, a sequence {un} is a PS c -sequence in X for I if I un 

c  ◦1 and I un  ◦1 strongly in X as n → ∞;

ii c ∈ R is a PS-value in X for I if there is a PS c -sequence in X for I;

iii I satisfies the PS c -condition in X if every PS c -sequence in X for I contains a

convergent subsequence;

iv I satisfies the PS-condition in X if for every c ∈ R, I satisfies the PS c-condition

in X.

Lemma 2.2 cf 6, Lemma 2.1 Under the assumption A1, the inner product

u, v1:



RN



∇u · ∇v  V−xuvdx 2.1

is well defined; therefore the corresponding norm u1 : u, u1 is well defined too, which is equivalent to the norm u  RN |∇u|2 u2dx1/2

.

Lemma 2.3 cf 8 Under the assumption that Vx ∈ L N/2RN  for the eigenvalue problem

−Δu  V−xu  μVxu, u ∈ E 2.2

there exists a sequence of eigenvalues μ n → ∞ such that the eigenfunction sequence ϕn is an orthonormal basis of E.

WhenPS c -condition is satisfied for all c∈ R, there are known methods of obtaining

an unbounded sequence of critical values of ϕsee, e.g., 9

Theorem 2.4 cf 10, Theorem 6.5 Suppose that E is an infinite-dimensional Banach space and

suppose ϕ ∈ C1E, R satisfies PS-condition, ϕu  ϕ−u for all u, and ϕ0  0 Suppose

E  E−⊕ E, where E−is finite dimensional, and assume the following conditions:

i there exist ζ > 0 and  > 0 such that if u   and u ∈ E, then ϕ u ≥ ζ;

ii for any finite-dimensional subspace W ⊂ E there exists R  RW such that ϕu ≤ 0 for

u ∈ W, u ≥ R.

Then ϕ possesses an unbounded sequence of critical values.

3 The PSc-Condition

Lemma 3.1 Under the assumptions A1, A2, and A3, for every c ∈ R, any PS c -sequence is bounded.

Proof By the eigenvalue problem inLemma 2.3, there exist k ∈ N such that eigenvalues are

μ1 < μ2 ≤ μ3 ≤ · · · ≤ μk ≤ λ < μk1 ≤ · · · for some λ ≥ 1; the corresponding eigenfunction

is ϕ1, ϕ2, ϕ3, , ϕk , ϕ k1, , then we denote X  X1

X2 , with X1k

i1span{ϕi}, X2 X⊥

1,

and denote u n ∈ X as un  vn  wn , where v n ∈ X1, w n ∈ X2 It’s obvious that



RN



|∇u|2 V−xu2− λVxu2

dx ≤ 0, ∀u ∈ X1, 3.1

and there exist δ > 0 such that



RN



|∇u|2 V−xu2− Vxu2

dx ≥ δu2

1, ∀u ∈ X2 3.2

byLemma 2.3 For any  > 0, there exists C  > 0 such that |Fx, u| ≥ C|u| α − |u|2fromA2 andA3 Choose 2 < α < α, then



RN

F x, undx − 1

α



RN

u n f x, undx

≤



RN



1− α

α



F x, undx

≤1− α

α



RN



C |un| α − |un|2

dx.

3.3

Let {un} be the sequence such that Iun → c, I un → 0 By inequality 3.2 and un 

v n  wn , v n ∈ X1, wn ∈ X2, and then

c  1  u1≥ Iun − 1

α

I un, un

 1 2



RN



|∇un|2− V xu2

n



dx−



RN

F x, undx

− 1

α



RN



|∇un|2− V xu2

n



dx 1

α



RN

u n f x, undx



 1

2 − 1

α



RN



|∇wn|2− V xw2

n  |∇vn|2− V xv2

n



dx

−



RN

F x, undx  1

α



RN

u n f x, undx

≥

 1

2 − 1

α



δ wn2

1

 1

2 − 1

α



vn2

1−

 1

2 − 1

α



RN



Vx|vn|2

dx

 α

α − 1

RN



C |un| α − |un|2

dx.

3.4

Boundary Value Problems 5

Choose  > 0 small, then for suitable C2, C3, the above inequality becomes

c  1  u1≥ C2un2

1 C3|un| α

α−

 1

2 − 1

α



|V|N/2 |vn|2

2 ∗. 3.5

Due to α > 2, it follows that {un} is bounded.

The following lemma is the same as6, Lemma 3.2 For the completeness, we prove it

Lemma 3.2 Under the assumptions A1, A2, A3, and A4, I satisfies the PS-condition in X.

Proof By Lemma 3.1, we know that any PS c sequence un is bounded in X Up to a subsequence, we may assume that u n u in X In order to establish strong convergence

it suffices to show

un1−→ u1. 3.6 SinceI un, un − u → 0, we infer that

0≤ lim sup

n→ ∞



un2

1− u2 1



 lim sup

n→ ∞ un , u n − u

 lim sup

n→ ∞



RN

f x, unun − udx.

3.7

We restrict our attention to the case N ≥ 3, but the cases N  1, 2 can be treated similarly Let

 > 0, for r≥ 1, then



|u n |≥r f x, unun − udx ≤ C4



|u n |≥r |un| p−1|un − u|dx

≤ C4r p−2 ∗

|u n |≥r |un|2∗−1|un − u|dx

≤ C4r p−2∗|un|2∗−1

2 ∗ |un − u|2 ∗.

3.8

Since p < 2∗, we may fix r large enough such that



|u n |≥r f x, unun − udx ≤ 

for all n Moreover, byA4 there exists R1> 0 such that



|u n |≤r ∩|x|≥R1f x, unun − udx ≤ |un|2|un − u|2 sup

|t|≤r,|x|≥R

f x, t

|t| ≤



3 3.10

for all n Finally, since un → u in L s BR10 for s ∈ 2, 2∗, we can use A2 again to derive



for n large enough Combining3.9–3.11 we conclude that



RN

f x, unun − udx ≤  3.12

for n large enough From this and3.7, we deduce 3.6 and complete the proof

4 Infinitely Many Solutions

We can obtain an infinite sequence of critical values fromTheorem 2.4

Proof of Theorem 1.1 We applyTheorem 2.4with E  X, ϕ  I It is clear that I ∈ C1X, R is

even because ofA1, A2, and A5 I0  0 By lemma 3.2, the PS-condition is satisfied.

From the proof ofLemma 3.1, we have X  X1

X2 , where X1  k

i1span{ϕi}, X2  X⊥

1

That is E−  X1, E X2 We only need to check conditionsi and ii

IntegratingA2, there is a constant C5 > 0 such that for all x∈ RN and t∈ R,

|Fx, t| ≤ C5



|t| p  |t| q

By the Sobolev embeding theorem and3.2, we have the estimate

I u ≥ 1

2



RN



|∇u|2 V−xu2

dx−1 2



RN

Vxu2dx − C5



RN



|u| p  |u| q

dx

≥ δ

2u2

1− C6u p

1− C7u q

1

4.2

for u ∈ X2 Letu1  and u ∈ X2,

I u ≥ δ

2

for small  Thus condition i is fulfilled with ζ  δ/22− C6 p − C7 q

ByA3, there is a constant C8such that|Fx, t| ≥ C8|t| α for every x∈ RN and |t| >  Indeed, let  > 0 small be given By integration ofA3, we have for x ∈ R N and|t| > ,

F x, t ≥ F x, 

 α |t| α ≥ C8|t| α 4.4

Boundary Value Problems 7

Let W be a finite-dimensional subspace of X Since all norms are equivalent of W and since

I u ≤ 1

2u2

1−1 2



RN

Vu2dx − C9u α

Also since α > 2, conditionii follows Thus we complete the proof

Acknowledgment

This work was supported by Key Program of NNSF of China10830005 and NNSF of China

10471024

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for all n Finally, since un → u in L s BR10 for s ∈ 2, 2∗, we can use A< small>2 again to... α ≥ C8|t| α 4.4

Boundary Value Problems 7

Let...

4 G Cerami, G Devillanova, and S Solimini, “Infinitely many bound states for some nonlinear scalar

field equations,” Calculus of Variations and Partial Di fferential Equations, vol