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Existence of solutions for semilinear elliptic equations on R N

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Abstract

In this paper, the existence of at least one nontrivial solution for a class of semilinear elliptic equations on R N is established by using the linking methods.

1 Introduction

In this paper we consider the question of the existence of solutions for a class of semilinear equations of the form

( P λ )u+λu=g(x,u),x R N ,

where λ>0 is a parameter and the nonlinearity gC( R N ×R) is asymptotically linear, i.e.,

lim | t | g ( x , t ) t =V(x), lim | x | V(x)= v
(1.1)

for some V(x)C( R N ,R) and v R. In case this equation is considered in a bounded domain Ω R N (with, say, the Dirichlet boundary condition), there is a large amount of literature on existence and multiplicity results, with the case of resonance being of particular interest (see [13]). We recall that the problem is said to be at resonance if λσ(S), where σ(S) denotes the spectrum of S, the ‘asymptotic linearization’ of the problem. In other words, S:D(S) L 2 (Ω) L 2 (Ω) is the operator given by

Su(x)=u(x)V(x)u(x),D(S)= H 0 1 (Ω) H 2 (Ω).
(1.2)

On the other hand, a systematic study of such asymptotically linear problems set in unbounded domains or the whole space R N is more recent and presents a number of mathematical difficulties (see [4, 5]). As an example, we note that in the case of problem ( P λ ), the asymptotic linearization operator S (now defined on D(S)= H 2 ( R N )) has a much more complicated spectrum (including an essential part [ v ,)), which in turn makes the study of this problem more challenging. In [4], motivated by the paper [5], Tehrani and Costa studied the existence of positive solutions to ( P λ ) by using the mountain pass theorem if g(x,u) satisfies some strong asymptotically linear conditions. Comparing with previous paper [4], in [6], Tehrani obtained the existence of a (possibly sign-changing) solution for problem ( P λ ) under essentially condition (1.1) only. In fact, he proved the following.

Theorem 1.0 [6]

Let g 0 (x,s):=g(x,s)V(x)s and assume that

  1. (G)

    for every ϵ>0, there exists 0 b ϵ (x) L 2 ( R N ) such that

    | g 0 ( x , s ) | b ϵ (x)+ϵ|s| a.e. x R N ,sR.

If Λ0 or max{0, v }<λ<Λ and λ σ p (S), then ( P λ ) has a solution in H 1 ( R N ).

Now, one naturally asks: Are there nontrivial solutions for problem ( P λ ) if λσ(S) in the above theorem? Obviously, this case is resonance. But, this problem is not easy because we face the difficulties of verifying that the energy functional satisfies the (PS) condition if we still follow the idea of [6]. Here, there is still an interesting problem: Are there nontrivial solutions for problem ( P λ ) if λσ(S) and g 0 (x,s) (in Theorem 1.0) is more generalized superlinear? We will answer the above problems affirmatively by using Li and Willem’s local linking methods (see [7]).

Next, we recall a few basic facts in the theory of Schrödinger operators which are relevant to our discussion (see [6]).

  1. 1.

    Since lim | x | V(x)= v , one has σ ess (S)=[ v ,).

  2. 2.

    The bottom of the spectrum σ(S) of the operator S is given by

    Λ= λ 0 = inf 0 u H 2 ( R N ) | u | 2 V ( x ) u 2 u 2 .

Therefore we clearly have Λ v . If Λ< v , then by using the concentration compactness principle of Lions, one shows that Λ is the principle eigenvalue of S with a positive eigenfunction Φ 0 :

S Φ 0 = λ 0 Φ 0 , Φ 0 H 2 ( R N ) , Φ 0 >0.
  1. 3.

    The spectrum of S in (, v ), namely σ(S)(, v ), is at most a countable set, which we denote by

    Λ= λ 0 < λ 1 < λ 2 <,

where each λ k is an isolated eigenvalue of S of the finite multiplicity. Let E λ j denote the eigenspace of S corresponding to the eigenvalue λ j .

Now, we state our main results. In this paper, we always assume that lim | x | V(x)= v and v <0. The conditions imposed on g 0 (x,t) (see Theorem 1.0) are as follows:

(H1) g 0 C( R N ×R,R), and there are constants C 1 , C 2 0 such that

| g 0 ( x , t ) | C 1 + C 2 | t | s 1 ,x R N ,tR,s ( 2 , p ) (N3),

where p = 2 N N 2 ;

(H2) g 0 (x,t)=(|t|), |t|0, uniformly on R N ;

(H3) lim | t | g 0 ( x , t ) t =+ uniformly on R N ;

(H4) There is a constant θ1 such that for all (x,t) R N ×R and s[0,1],

θ ( g 0 ( x , t ) t 2 G 0 ( x , t ) ) ( s g 0 ( x , s t ) t 2 G 0 ( x , s t ) ) ,

where G 0 (x,t)= 0 t g 0 (x,s)ds;

(H5) For some δ>0, either

G 0 (x,t)0for |t|δ,x R N

or

G 0 (x,t)0for |t|δ,x R N ;

(H6) lim | x | sup | t | r g 0 ( x , t ) | t | =0 for every r>0.

Theorem 1.1 Assume that conditions (H1)-(H4) hold. Ifλ is an eigenvalue of S(λ< v ), assume also that (H5) and (H6) hold. Then the problem ( P λ ) has at least one nontrivial solution.

Remark 1.1 It follows from the condition (H3) that our nonlinearity g 0 (x,t) does not satisfy the classical condition of Ambrosetti and Rabinowitz:

(AR) There is μ>2 such that 0<μ G 0 (x,u)u g 0 (x,u) for all x R N and u0.

In recent years, there have been some papers devoted to replacing (AR) with more natural conditions (see [810]). But our methods are different from the references therein.

We also consider asymptotically quadratic functions. We assume that:

(H7) For every ϵ>0, there exists 0 b ϵ (x) L 2 ( R N ) such that

| g 0 ( x , s ) | b ϵ (x)+ϵ|s|a.e. x R N ,sR,

and λ k <λ< λ k + 1 .

Theorem 1.2 Assume that conditions (H2), (H6), (H7) and one of the following conditions hold:

(A1) λ j <0< λ j + 1 , jk;

(A2) λ j =0< λ j + 1 , jk for some δ>0,

G 0 (x,u)0 for |u|>δ,x R N ;

(A3) λ j <0= λ j + 1 , jk for some δ>0,

G 0 (x,u)0 for |u|δ,x R N .

Then problem ( P λ ) has at least one nontrivial solution.

2 Preliminaries

Let X be a Banach space with a direct sum decomposition

X= X 1 X 2 .

Consider two sequences of subspaces

X 0 1 X 1 1 X 1 , X 0 2 X 1 2 X 2

such that

X j = n N X n j ,j=1,2.

For every multi-index α=( α 1 , α 2 ) N 2 , let X α = X α 1 X α 2 . We know that

αβ α 1 β 1 , α 2 β 2 .

A sequence ( α n ) N 2 is admissible if, for every α N 2 , there is mN such that nm α n α. For every I:XR, we denote by I α the function I restricted X α .

Definition 2.1 Let I be locally Lipschitz on X and cR. The functional I satisfies the ( C ) c condition if every sequence ( u α n ) such that ( α n ) is admissible and

u α n X α n ,I( u α n )c, ( 1 + u α n ) I ( u α n )0

contains a subsequence which converges to a critical point of I.

Definition 2.2 Let I be locally Lipschitz on X and cR. The functional I satisfies the ( C ) condition if every sequence ( u α n ) such that ( α n ) is admissible and

u α n X α n , sup n I( u α n )c, ( 1 + u α n ) I ( u α n )0

contains a subsequence which converges to a critical point of I.

Remark 2.1 1. The ( C ) condition implies the ( C ) c condition for every cR.

  1. 2.

    When the ( C ) c sequence is bounded, then the sequence is a ( PS ) c sequence (see [11]).

  2. 3.

    Without loss of generality, we assume that the norm in X satisfies

    u 1 + u 2 2 = u 1 2 + u 2 2 , u j X j ,j=1,2.

Definition 2.3 Let X be a Banach space with a direct sum decomposition

X= X 1 X 2 .

The function I C 1 (X,R) has a local linking at 0, with respect to ( X 1 , X 2 ) if, for some r>0,

I ( u ) 0 , u X 1 , u r , I ( u ) 0 , u X 2 , u r .

Lemma 2.1 (see [7])

Suppose that I C 1 (X,R) satisfies the following assumptions:

(B1) I has a local linking at 0 and X 1 {0};

(B2) I satisfies ( PS ) ;

(B3) I maps bounded sets into bounded sets;

(B4) For every mN, I(u), u, uX= X m 1 X 2 . Then I has at least two critical points.

Remark 2.2 Assume that I satisfies the ( C ) c condition. Then this theorem still holds.

Let X be a real Hilbert space and let I C 1 (X,R). The gradient of I has the form

I(u)=Au+B(u),

where A is a bounded self-adjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.

We assume that there exist an orthogonal decomposition,

X= X 1 + X 2 ,

and two sequences of finite-dimensional subspaces,

X 0 1 X 1 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 ,

such that

X j = n N X n j ¯ , j = 1 , 2 , A X n j X n j , j = 1 , 2 , n N .

For every multi-index α=( α 1 , α 2 ) N 2 , we denote by X α the space

X α 1 X α 2 ,

by p α :X X α the orthogonal projector onto X α , and by M (L) the Morse index of a self-adjoint operator L.

Lemma 2.2 (see [7])

I satisfies the following assumptions:

  1. (i)

    I has a local linking at 0 with respect to ( X 1 , X 2 );

  2. (ii)

    There exists a compact self-adjoint operator B such that

    B(u)= B (u)+ ( u ) ,u;
  3. (iii)

    A+ B is invertible;

  4. (vi)

    For infinitely many multiple-indices α:=(n,n),

    M ( ( A + P α B ) | X α ) dim X n 2 .

Then I has at least two critical points.

3 The proof of main results

Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional

I(u)= 1 2 ( | u | 2 V ( x ) | u | 2 ) + 1 2 λ u 2 Ω G 0 (x,u)

defined on X= H 1 ( R N ). We consider only the case λσ(S), and

G 0 (x,u)0for |u|δ,x R N .
(3.1)

Then other case is similar and simple.

Let X 2 be a finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of S+λ and let X 1 be its orthogonal complement in X. Choose a Hilbertian basis e n (n0) for X and define

X n 1 = span ( e 0 , e 1 , , e n ) , n N ; X n 2 = X 2 , n N ; X 1 = n N X n 1 ¯ .

By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to C 1 (X,R) and maps bounded sets to bounded sets.

  1. (2)

    We claim that I has a local linking at 0 with respect to ( X 1 , X 2 ). Decompose X 1 into V+W when V= E λ , W= ( X 2 + V ) . Also, set u=v+w, u X 1 , vV, wW.

For the convenience of our proof, we state some facts for the norm of the whole space X. It is well known that there is an equivalent norm on X= H 1 ( R N ) such that

( | u | 2 V ( x ) | u | 2 ) = u 2 ,u X 2

and

( | u | 2 V ( x ) | u | 2 ) = u 2 ,uW.

By the equivalence of norm in the finite-dimensional space, there exists C>0 such that

v Cv,vV.
(3.2)

It follows from (H1) and (H2) that for any ϵ>0, there exists C ϵ such that

| G 0 ( x , u ) | ϵ u 2 + C ϵ | u | s .
(3.3)

Hence, we obtain

I(u)m u 2 + c u s + 1 ,

where m>0, c is a constant and hence, for r>0 small enough,

I(u)0,u X 2 , u X r.

Let u=v+w X 1 be such that u X r 1 = δ 2 C and let

A 1 = { x R N : | w ( x ) | δ 2 } , A 2 = R N A 1 .

From (3.2), we have

| v ( x ) | v Cv δ 2

for all u r 1 and x R N . On the one hand, one has |u(x)||v(x)|+|w(x)| v + δ 2 δ for all x A 1 . Hence, from (H5), we obtain

A 1 G 0 (x,u)dx0.

On the other hand, we have

| u ( x ) | | v ( x ) | + | w ( x ) | δ 2 + | w ( x ) | 2 | w ( x ) |

for all x A 2 . It follows from (3.3) that

G 0 (x,u)ϵ u 2 + C ϵ | u | s + 1 4ϵ w 2 + 2 s + 1 C ϵ | w | s + 1

for all x A 2 and all u X 1 with u r 1 , which implies that

G 0 ( x , u ) 4 ϵ A 2 w 2 d x + A 2 2 s + 1 C ϵ | w | s + 1 d x 4 ( C 3 ) 2 ϵ w 2 + ( 2 C 3 ) λ + 1 C ϵ w s + 1 ,

where C 3 is a constant. Hence, there exist positive constants C , C 4 and C 5 such that

I ( u ) = 1 2 w 2 A 2 G 0 ( x , u ) d x A 1 G 0 ( x , u ) d x C w 2 4 ( C 3 ) 2 ϵ w 2 ( 2 C 3 ) λ + 1 C ϵ w s + 1 A 1 G ( x , u ) d x C 4 w 2 C 5 w s + 1

for all u X 1 with u r 1 , which implies that

I(u)0,u X 1  with ur

for 0<r small enough.

  1. (3)

    We claim that I satisfies ( C ) c . Consider a sequence ( u α n ) such that ( u α n ) is admissible and

    u α n X α n ,I( u α n )c, ( 1 + u α n ) I ( u α n )0
    (3.4)

and

lim n ( 1 2 g 0 ( x , u α n ) u α n G 0 ( x , u α n ) ) =c.
(3.5)

Let w α n = u α n 1 u α n . Up to a subsequence, we have

w α n win X, w α n win  L loc 2 , w α n (x)w(x)a.e. x R N .

If w=0, we choose a sequence { t n }[0,1] such that

I( t n u α n )= max t [ 0 , 1 ] I(t u α n ).

For any m>0, let v α n =2 m w α n . Now, we claim that

lim n G 0 (x, v α n )=0.

Let ϵ>0; for r1, then,

| v α n | r G 0 ( x , v α n ) d x C 6 r p 2 | v α n | r | v α n | 2 d x C 7 r p 2 | v α n | 2 2 .

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

| | v α n | r G 0 ( x , v α n ) d x | ϵ 3

for all n. Moreover, by (H6), there exists R>0 such that

| | v α n | r G 0 ( x , v α n ) d x | | v α n | 2 2 sup | t | r , | x | R | G 0 ( x , t ) | t 2 ϵ 3

for all n. Finally, since v α n 0 in L s ( B R (0)) for s[2, 2 ), we can use (H1) again to derive

| | v α n | r | x | R G 0 ( x , v α n ) d x | ϵ 3

for n large enough. Combining the above three formulas, our claim holds.

So, for n large enough, 2 m u α n 1 (0,1), we have

I( t n u α n )I( v α n )mϵ m 2 ,
(3.6)

where ϵ is a small enough constant.

That is, I( t n u α n ). Now, I(0)=0, I( u α n )c, we know that t n [0,1] and

( | ( t n u α n ) | 2 V ( x ) t n 2 | u α n | 2 ) + λ t n 2 | u α n | 2 g 0 ( x , t n u α n ) t n u α n = t n d d t | t = t n I ( t u α n ) = 0 .
(3.7)

Therefore, using (H4), we have

1 2 g 0 (x, u α n ) u α n G 0 (x, u α n ) 1 θ ( 1 2 g 0 ( x , t n u α n ) t n u α n G 0 ( x , t n u α n ) ) +.

This contradicts (3.5).

If w0, then the set ={x R N :w(x)0} has a positive Lebesgue measure. For x, we have | u α n (x)|. Hence, by (H3), we have

g 0 ( x , u α n ( x ) ) u α n ( x ) | u α n ( x ) | 2 | w α n ( x ) | 2 .
(3.8)

From (3.4), we obtain

1(1) ( w 0 + w = 0 ) g 0 ( x , u α n ( x ) ) u α n ( x ) | u α n ( x ) | 2 | w α n ( x ) | 2 dx.
(3.9)

By (3.8), the right-hand side of (3.9) +. This is a contradiction.

In any case, we obtain a contradiction. Therefore, { u α n } is bounded.

Next, we denote { u α n } as { u n } and prove { u n } contains a convergent subsequence.

In fact, we know that { u n } is bounded in X. Passing to a subsequence, we may assume that u n u in X. In order to establish strong convergence, it suffices to show that

u n u.

By the condition (H6) and I ( u n ), u n u0, we can similarly conclude it according to the above proof of our claim.

Finally, we claim that for every mN,

I(u)as u,u X m 1 X 2 .

By (H2) and (H3), there exist large enough M and some positive constant T such that

G 0 (x,t)M t 2 ,x R N ,tT.

So, for any u X m 1 X 2 , we have

I ( t u ) = 1 2 t 2 ( | u | 2 V ( x ) | u | 2 ) + t 2 2 λ u 2 G 0 ( x , t u ) 1 2 t 2 ( | u | 2 V ( x ) | u | 2 ) + t 2 2 λ u 2 M t 2 u 2 as  t + .

Hence, our claim holds. □

Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one since our result is a variant of Ding Yanheng’s Theorem 1.2 (see [12]). □

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Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.

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Correspondence to Ruichang Pei.

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Keywords

  • Schrödinger equation
  • subcritical exponent
  • local linking