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

Boundary Value Problems

Volume 2009, Article ID 532546,7pages 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,[email protected] Received 23 March 2009; Accepted 10 June 2009

Recommended by Martin Schechter

We consider a similinear elliptic equation with sign-changing potential−Δu−Vxu fx, u, u∈H¹R^N, whereVxis a function possibly changing sign inR^N. Under certain assumptions onf, we prove that the equation has infinitely many solutions.

Copyrightq2009 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−Vxufx, u, u∈H¹ R^N

P is studied, whereVxis a function possibly changing sign,f is a continuous function on R^N×R.

ProblemParises 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 problemPwithVx ≡ λ λis a positive constant and fx, u |u|^p−2u. And Grossi et al. 3 established some existence results for −Δu λuaxgu, whereaxis a function possibly changing sign,guhas superlinear growth and λ is a positive real parameter; he discussed both the cases of subcritical and critical growth forguand proved the existence of linking type solutions.

Cerami et al. 4 prove that the problem P has infinitely many solutions, where ax is a regular function such that lim inf_{|x| → ∞}ax a_∞ > 0 and some suitable decay assumptions, fx, u |u|^p−2u. Kryszewski and Szulkin 5 considered the existence of

a nontrivial solution of P in a situation where fx, u and Vx are periodic in the x- variable, fx, uis superlinear at u 0 and ±∞, and 0 lies in a spectral gap of−ΔuV. If in additionfx, uis odd inu,Phas infinitely many solutions.

In6, Zeng and Li proved existence ofm−npairs of nontrivial solutionsm > n, m andnare integersofP, under the assumption thatVxis a function possibly changing sign inR^Nandfx, usatisfies some growth conditions.

In this paper, we prove the existence of infinitely many solutions ofP, under the assumption that Vxis a function possibly changing sign inR^N andfx, ualso satisfies some growth conditions. One difficulty in considering problemPis the loss of compactness because ofR^N; the other is thatVxmay 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 fixedi}.Vx Vx−V⁻x, where V^±max{±Vx,0}.Ω1{x∈R^N;V⁻x/0},Ω2{x∈R^N;V⁻x 0}.

Xis defined as the completion ofDR^Nwith respect to the inner product

u, v₁:

R^N

∇u· ∇vV⁻xuv

dx. 1.1

The functional associated withPis Iu: 1

2

R^N|∇u|²V⁻xu²dx−1 2

R^NVxu²dx−

R^NFx, udx, 1.2 foru∈X, whereFx, u _u

0fx, tdt.

Our fundamental assumptions are as follows:

A1Vx ∈ L^N/2R^N, meas{x ∈R^N;Vx/0} > 0.V⁻x ∈L^∞R^N,Ω2 is a strip region, lim_{|x| → ∞}V⁻x a >0 inΩ1.

A2f ∈ CR^N ×Rand there are constants C₁ > 0 and 2 < p ≤ q < 2^∗ such that

|fx, t| ≤C₁|t|^p−1|t|^q−1.

A3There existsα >2 such that 0< αFx, t≤tfx, tfor everyx∈R^Nandt /0.

A4lim_{|x| → ∞}sup_|t|≤r|fx, t|/|t| 0 for everyr >0.

A5For anyt∈R, fx, t −fx,−t.

Here 2^∗denotes the critical Sobolev exponent, that is, 2^∗ 2N/N−2forN≥3 and 2^∗∞forN1,2.

Theorem 1.1. Under the assumptionsA1–A5,Ppossesses infinitely many solutions onX.

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

ki1aix|t|^pi−2t with 2 < pi < 2^∗ and for i 1, . . . , k, the nonnegative function aix ∈ L^∞R^N,lim_{|x| → ∞}a_ix 0.

2. Preliminaries

We define the Palais-Smaledenoted byP Ssequences,P S-values, andP S-conditions inXforIas follows.

Definition 2.1cf.7. iForc∈R, a sequence{un}is aP S_c-sequence inXforIifIun

c◦1andIun ◦1strongly inXasn → ∞;

iic∈Ris aP S-value inXforIif there is aP S_c-sequence inXforI;

iiiI satisfies theP S_c-condition inXif everyP S_c-sequence inX forI contains a convergent subsequence;

ivIsatisfies theP S-condition inXif for everyc∈R,Isatisfies theP S_c-condition inX.

Lemma 2.2cf.6, Lemma 2.1. Under the assumptionA1, the inner product u, v₁:

R^N

∇u· ∇vV⁻xuv

dx 2.1

is well defined; therefore the corresponding norm u₁ :

u, u₁ is well defined too, which is equivalent to the normu

R^N|∇u|²u²dx^1/2.

Lemma 2.3cf.8. Under the assumption thatVx∈L^N/2R^Nfor the eigenvalue problem

−ΔuV⁻xuμVxu, u∈E 2.2

there exists a sequence of eigenvalues μ_n → ∞ such that the eigenfunction sequence ϕ_n is an orthonormal basis ofE.

WhenP S_c-condition is satisfied for allc∈R, there are known methods of obtaining an unbounded sequence of critical values ofϕsee, e.g.,9.

Theorem 2.4cf.10, Theorem 6.5. Suppose thatEis an infinite-dimensional Banach space and suppose ϕ ∈ C¹E,Rsatisfies P S-condition, ϕu ϕ−ufor all u, andϕ0 0. Suppose EE⁻⊕E, whereE⁻is finite dimensional, and assume the following conditions:

ithere existζ >0 and >0 such that ifuandu∈E, thenϕu≥ζ;

iifor any finite-dimensional subspaceW ⊂Ethere existsRRWsuch thatϕu≤0 for u∈W,u ≥R.

Thenϕpossesses an unbounded sequence of critical values.

3. The PS

-Condition

Lemma 3.1. Under the assumptionsA1,A2, andA3, for everyc∈R, anyP S_c-sequence is bounded.

Proof. By the eigenvalue problem inLemma 2.3, there existk ∈Nsuch that eigenvalues are μ₁ < μ₂ ≤ μ₃ ≤ · · · ≤ μ_k ≤ λ < μ_k1 ≤ · · · for someλ ≥ 1; the corresponding eigenfunction

isϕ1, ϕ2, ϕ3, . . . , ϕk, ϕ_k1, . . ., then we denoteXX1

X2, withX1_k

i1span{ϕi}, X2X₁^⊥, and denoteu_n∈X asu_nv_nw_n, wherev_n∈X₁, w_n ∈X₂. It’s obvious that

R^N

|∇u|²V⁻xu²−λVxu²

dx≤0, ∀u∈X1, 3.1

and there existδ >0 such that

R^N

|∇u|²V⁻xu²−Vxu²

dx≥δu²₁, ∀u∈X₂ 3.2

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

R^NFx, undx− 1 α

R^Nu_nfx, undx

≤

R^N

1− α

α

Fx, undx

≤ 1− α

α

R^N

C|un|^α−|un|² dx.

3.3

Let {un} be the sequence such that Iun → c, Iun → 0. By inequality 3.2 and un vnwn, vn∈X1, wn∈X2, and then

c1u₁≥Iun− 1 α

Iun, un

1

2

R^N

|∇un|²−Vxu²_n dx−

R^NFx, undx

− 1 α

R^N

|∇un|²−Vxu²_n dx 1

α

R^Nu_nfx, undx

1 2 − 1

α

R^N

|∇wn|²−Vxw²_n|∇vn|²−Vxv²_n dx

−

R^NFx, undx 1 α

R^Nu_nfx, undx

≥ 1

2 − 1 α

δwn²₁ 1

2 − 1 α

vn²₁−

1 2 − 1

α

R^N

Vx|vn|² dx

α α−1

R^N

C|un|^α−|un|² dx.

3.4

Choose >0 small, then for suitableC2, C3, the above inequality becomes c1u₁≥C₂un²₁C₃|un|^α_α−

1 2 − 1

α

|V|_N/2|vn|²₂∗. 3.5

Due toα >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 assumptionsA1,A2,A3, andA4,Isatisfies theP S-condition inX.

Proof. By Lemma 3.1, we know that any P S_c sequence un is bounded in X. Up to a subsequence, we may assume thatu_n u inX. In order to establish strong convergence it suffices to show

un₁−→ u₁. 3.6

SinceIun, un−u → 0, we infer that 0≤lim sup

n→ ∞

un²₁− u²₁

lim sup

n→ ∞ un, un−u lim sup

n→ ∞

R^Nfx, unun−udx.

3.7

We restrict our attention to the caseN≥3, but the casesN1,2 can be treated similarly. Let >0, forr ≥1, then

|un|≥rfx, unun−udx≤C₄

|un|≥r|un|^p−1|un−u|dx

≤C₄r^p−2∗

|un|≥r|un|^2∗−1|un−u|dx

≤C₄r^p−2∗|un|²₂^∗∗⁻¹|un−u|₂∗.

3.8

Sincep <2^∗, we may fixrlarge enough such that

|un|≥rfx, unun−udx≤

3 3.9

for alln. Moreover, byA4there existsR₁>0 such that

|un|≤r∩|x|≥R1fx, unun−udx≤ |u_n|₂|un−u|₂ sup

|t|≤r,|x|≥R1

fx, t

|t| ≤

3 3.10

for alln. Finally, sinceun → uinL^sBR10fors∈2,2^∗, we can useA2again to derive

|un|≤r∩|x|≤R1fx, unun−udx≤

3 3.11

fornlarge enough. Combining3.9–3.11we conclude that

R^Nfx, unun−udx≤ 3.12

fornlarge enough. From this and3.7, we deduce3.6and complete the proof.

4. Infinitely Many Solutions

We can obtain an infinite sequence of critical values fromTheorem 2.4.

Proof ofTheorem 1.1. We applyTheorem 2.4withEX, ϕ I. It is clear thatI ∈C¹X,Ris even because ofA1,A2, andA5.I0 0. By lemma 3.2, theP S-condition is satisfied.

From the proof ofLemma 3.1, we haveX X1

X2, whereX1 _k

i1span{ϕi}, X2 X₁^⊥. That isE⁻ X₁, EX₂. We only need to check conditionsiandii.

IntegratingA2, there is a constantC₅ >0 such that for allx∈R^Nandt∈R,

|Fx, t| ≤C5

|t|^p|t|^q

. 4.1

By the Sobolev embeding theorem and3.2, we have the estimate Iu≥ 1

2

R^N

|∇u|²V⁻xu² dx−1

2

R^NVxu²dx−C₅

R^N

|u|^p|u|^q dx

≥ δ

2u²₁−C₆u^p₁−C₇u^q₁

4.2

foru∈X2. Letu₁andu∈X2, Iu≥ δ

2²−C₆^p−C₇^q>0 4.3

for small. Thus conditioniis fulfilled withζ δ/2²−C₆^p−C₇^q.

ByA3, there is a constantC8such that|Fx, t| ≥ C8|t|^αfor everyx∈R^N and |t|> . Indeed, let >0 small be given. By integration ofA3, we have forx∈R^N and|t|> ,

Fx, t≥ Fx,

^α |t|^α≥C8|t|^α. 4.4

LetWbe a finite-dimensional subspace ofX. Since all norms are equivalent ofWand since Iu≤ 1

2u²₁−1 2

R^NVu²dx−C9u^α_α. 4.5

Also sinceα >2, conditioniifollows. Thus we complete the proof.

Acknowledgment

This work was supported by Key Program of NNSF of China10830005and NNSF of China 10471024.

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