AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2021.126034AM-110161ArticlesPhysics&Mathematics Sign Changing Solution of a Semilinear Schr&#246;dinger Equation with Constraint JingkunLiu1*QiFan2Chengyi University College, Jimei University, Xiamen, ChinaXiamen Sinic-Tek Intelligent Technology Co., Ltd., Xiamen, China23062021120648949914, April 202125, June 2021 28, June 2021© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

The purpose of this paper is to study a semilinear Schr &#246;dinger equation with constraint in H 1( R N), and prove the existence of sign changing solution. Under suitable conditions, we obtain a negative solution, a positive solution and a sign changing solution by using variational methods.

Semilinear Schr&#246;dinger Equation Sign Changing Solution Palais-Smale Condition Pseudo-Gradient Vector Field
1. Introduction

{ − Δ u + V ( x ) u = λ f ( x , u ) ,       x ∈   R N ,     u ∈ H 1 ( R N ) , ∫ R N ( | ∇ u | 2 + V ( x ) u 2 ) d x = r 2 , u ( x ) → 0,           | x | → + ∞ . (1.1)

Given r > 0 , we try to find ( u , λ ) to satisfy the Equation (1.1). We say ( u , λ ) a positive solution if u is positive, a negative solution if u is negative, and a sign changing solution if u is sign changing.

Several authors have considered a Schrödinger equation of the form

{ − Δ u + V ( x ) u = f ( x , u ) ,       x ∈   R N ,     u ∈ H 1 ( R N ) , u ( x ) → 0,           | x | → + ∞ . (1.2)

In Bartsch and Wang , it is shown that the problem (1.2) possesses infinitely many solutions when f ( x , u ) is odd with respect to u. Liu  obtains a positive solution and a negative solution of the problem (1.2) under the assumption that V ( x ) and f ( x , u ) are periodic with respect to the x-variables. Bartsch, Liu and Weth  prove the existence of sign changing solutions to the problem (1.2) and estimate the number of nodal domain.

Some papers concern with the problem (1.1). Under some conditions, a positive and a negative solution can be found in  and .  gives some results on the existence of sign changing and multiple solutions of the problem (1.1) with different conditions.

In order to state our results, we require the following assumptions:

(A1) V ∈ L l o c ∞ ( R N ) ,   inf x ∈ R N V ( x ) > 0 .

(A2) f : R N × R → R is locally Lipschitz continuous, and there are constants C > 0 and p ∈ ( 2,   2 * ) such that

| f ( x , t ) | ≤ C ( 1 + | t | p − 1 )     for   x ∈ R N ,   t ∈ R ,

where 2 * : = 2 N N − 2 for N ≥ 3 and 2 * : = ∞ for N = 1 , 2 . Moreover, f ( x , t ) = o ( | t | ) as t → 0 uniformly in x.

(A3) There is a constant η > 2 such that

0 ≤ η F ( x , t ) ≤ f ( x , t ) t     for   x ∈ R N ,     t ∈ R ,

where F ( x , t ) : = ∫ 0 t     f ( x , s ) d s for x ∈ R N ,   t ∈ R .

(A4) There is an open subset Ω ⊂ R N such that t f ( x , t ) > 0 for x ∈ Ω and | t | sufficiently large.

(A5) l i m | x | → + ∞ sup | t | ≤ r | f ( x , t ) | | t | = 0 for every r > 0 .

Our main result is the following theorem.

Theorem 1.1 Suppose (A1)-(A5) hold. Then problem (1.1) has at least three nontrivial solutions u + ,   u − , and u ¯ , where u + is positive, and u − is negative and u ¯ changes sign.

The key point is to construct certain invariant sets of the gradient flow associated with the energy functional of the elliptic problem. All positive and negative solutions are contained in these invariant sets. And minimax procedures can be used to construct sign changing critical point of the energy functional outside these invariant sets.

2. Preliminaries

We first fix some notations. Denote the usual Sobolev space by W m ,   p ( R N ) , and set H m ( R N ) = W m ,   2 ( R N ) . Consider the Hilbert space

H : = { u ∈ H 1 ( R N ) :   ∫ R N     V ( x ) u 2 d x < + ∞ } .

We introduce the inner product in H by the formula

〈 u , v 〉 : = ∫ R N ( ∇ u ⋅ ∇ v + V ( x ) u v ) d x     for   u , v ∈ H ,

and the corresponding norm

‖ u ‖ : = 〈 u , u 〉     for   u ∈ H .

According to (A1), there is a continuous embedding H ↪ H 1 ( R N ) , hence

H ↪ L s ( R N )     for     2 ≤ s ≤ 2 * . (2.1)

Note that by (A2) for any ε > 0 , there is a constant K 1 ( ε ) > 0 such that

| f ( x , t ) | ≤ ε | t | + K 1 ( ε ) | t | p − 1     for     x ∈ R N ,   t ∈ R . (2.2)

Assumption (A3) implies that given δ > 0 , there exists a constant K 2 ( δ ) > 0 such that

F ( x , t ) ≥ K 2 ( δ ) | t | η − δ | t | 2     for     x ∈ R N ,   t ∈ R . (2.3)

Denote

S r : = { u ∈ H : ∫ R N ( | ∇ u | 2 + V ( x ) u 2 ) d x = r 2 } ,

F ( x , u ) : = ∫ 0 u     f ( x , t ) d t       for   u ∈ R , (2.4)

J ( u ) : = − ∫ R N   F ( x , u ) d x       for     u ∈ H , (2.5)

I : = J | S r . (2.6)

By Zeidler , we have

I ′ ( u ) = J ′ ( u ) − 〈 J ′ ( u ) , u 〉 ‖ u ‖ 2 u       for   u ∈ S r , (2.7)

where 〈 J ′ ( u ) , v 〉 = − ∫ R N   f ( x , u ) v d x for u ,   v ∈ H . It is easy to see from (2.7) that the critical points of I correspond to the solutions of problem (1.1) with λ = − ‖ u ‖ 2 〈 J ′ ( u ) , u 〉 . And I is bounded.

Definition 2.1 Suppose E is a real Banach space. For Φ ∈ C 1 ( E , R ) , we say Φ satisfies the Palais-Smale condition (denoted by (PS)) if any sequence { u n } ⊂ E for which { Φ ( u n ) } is bounded and Φ ′ ( u n ) → 0 possesses a convergent subsequence. We say Φ satisfies (PS)c for a fixed c ∈ R if any sequence { u n } ⊂ E for which Φ ( u n ) → c and Φ ′ ( u n ) → 0 possesses a convergent subsequence. We say Φ satisfies (PS)+ if Φ satisfies (PS)c for all c > 0 ; Φ satisfies (PS) if I satisfies (PS)c for all c < 0 .

Lemma 2.1  I satisfies (PC).

Let G be the Nemytskii operator induced by f, the mapping I ′ may be written as

I ′ ( u ) = − T ( u ) u − K G ( u )       for   u ∈ S r , (2.8)

where T ( u ) : = − 1 r 2 ∫ R N     f ( x , u ) u d x and K : = ( − Δ + V ) − 1 . Note that

K G ( u ) = ( − Δ + V ) − 1 f ( ⋅ ,   u ( ⋅ ) )       for   u ∈ H .

In other words, KG is uniquely determined by the relation

〈 K G ( u ) ,   v 〉 = ∫ R N   f ( x , u ) v d x ,       for   u ,   v ∈ H . (2.9)

I ′ ( u ) is globally Lipschitz continuous in H applying (A2) .

Let E be a real Banach space, Φ ∈ C 1 ( E , R ) and E ¯ : = { u ∈ E :   Φ ′ ( u ) ≠ 0 } . We will give some relevant definitions below.

Definition 2.2 A locally Lipschitz continuous mapping Q :   E ¯ → E is called a pseudo-gradient vector field (denoted by p.g.v.f) for Φ on E ¯ if it satisfies the following conditions

1) ‖ Q ( u ) ‖ E ≤ 2 ‖ Φ ′ ( u ) ‖ E * ;

2) 〈 Φ ′ ( u ) , Q ( u ) 〉 E ≥ ‖ Φ ′ ( u ) ‖ E * 2 .

Suppose Q is a p.g.v.f for Φ on E ¯ , and consider the initial value problem in E ¯

{ d d t σ ( t , u ) = − Q ( σ ( t , u ) ) ,       t ≥ 0 σ ( 0 , u ) = u . (2.10)

According to the theory of ordinary differential equations in Banach space , (2.10) has a unique solution in E ¯ , denoted by σ ( t , u ) , with right maximal interval of existence [ 0, ω + ( u ) ) . Note that ω + ( u ) may be either a positive number or + ∞ . Note also that Φ ( σ ( t , u ) ) is monotonically decreasing on [ 0, ω + ( u ) ) and therefore σ ( t , u )   ( 0 ≤ t < ω + ( u ) ) is called a descending flow curve.

Definition 2.3 A nonempty subset M of E is called an invariant set of descending flow for Φ determined by Q if

{ σ ( t , u ) | 0 ≤ t < ω + ( u ) } ⊂ M

for all u ∈ E ¯ ∩ M .

Definition 2.4 Let M and D be invariant sets of descending flow for Φ ,   D ⊂ M . Denote

C M ( D ) = { u | u ∈ D , or   u ∈ M \ D   andthereis   0 ≤ t < ω + ( u )   suchthat   σ ( t , u ) ∈ D }

If D = C M ( D ) , then D is called a complete invariant set of descending flow relative to M.

3. Invariant Subsets of the Descending Flow

In this section we shall recall some results about the flow generated by I ′ . We refer to Mawhin and Willem  for details.

It is clear that I ′ is globally Lipschitz continuous, and I ′ is a p.g.v.f of I. In the following we consider the initial value problem

{ d σ d t = − I ′ ( σ ) = T ( σ ) σ + K G ( σ ) ,           t ≥ 0 , σ ( 0 ,   u ) = u ,                   u ∈ S r . (3.1)

Applying the theory of ordinary differential equations, we obtain:

Lemma 3.1  There exists a unique solution t ↦ σ ( t , u ) of (3.1) defined on a maximal interval [ 0, ω + ( u ) ) with 0 < ω + ( u ) ≤ + ∞ . The flow σ :   D ↦ S r is continuous, where D : = { ( t ,   u ) :   0 ≤ t < ω + ( u ) ,   u ∈ S r } . For 0 ≤ t < ω + ( u ) ,   σ has the expression

σ ( t ,   u ) = e ∫ 0 t     T ( σ ( s ,   u ) ) d s ( u + ∫ 0 t     e − ∫ 0 s     T ( σ ( ξ ,   u ) ) d ξ K G ( σ ( s ,   u ) ) d s ) . (3.2)

Lemma 3.2  If ω + ( u ) is finite, then I ( σ ( t ,   u ) ) → − ∞ as t → ω + ( u ) .

In our case, I is bounded and so it follows from Lemma 3.2 that ω + ( u ) = + ∞ for u ∈ S r .

Lemma 3.3  Suppose c < b < 0 , for any u ∈ I − 1 ( [ c ,   b ] ) , either there exists a unique t ( u ) ∈ [ 0,   + ∞ ) such that I ( σ ( t ( u ) , u ) ) = c or there is a critical point v of I in I − 1 ( [ c ,   b ] ) , such that σ ( t ,   u ) → v as t → + ∞ .

It is easy to verify that d ( ‖ σ ‖ 2 ) d t = 0 , that is

σ ( t ,   u ) ∈ S r     for   u ∈ S r . (3.3)

In our further proof, we shall need the following Lemma which is derived by Brézis and extended by Martin to infinite dimensional space (cf. Theorem 1.6.3 in  ).

Lemma 3.4  Suppose E is a real Banach space, D is a closed subset of E, Q :   E → E is locally Lipschitz continuous and

lim h ↓ 0 d i s t E ( u + h Q ( u ) ,   D ) h = 0     for     u ∈ ∂ D . (3.4)

where d i s t E ( .   ,   . ) is the distance on E. If u 0 ∈ D and σ ( t ) with 0 ≤ t < ω + ( u 0 ) is the solution of the initial value problem

{ d σ d t = Q ( σ ) , σ ( 0 ,   u 0 ) = u 0 ,

then σ ( t ) ∈ D for all t ∈ [ 0, ω + ( u 0 ) ) .

Next we will discuss the convex cones P + : = { u ∈ H :   u ≥ 0 } , and P − : = { u ∈ H :   u ≤ 0 } . Moreover, for u ∈ H we denote that u + : = max { u , 0 } and u − : = min { u , 0 } . Note that u ∈ H implies u ± ∈ P ± . Consider the sets

P ε + : = { u ∈ H : d i s t ( u , P + ) < ε } ,       P ε − : = { u ∈ H : d i s t ( u , P − ) < ε } ,

as well as P ε : = P ε + ¯ ∪ P ε − ¯ for ε > 0 . Note that P ε + and P ε − are open convex subsets of H, whereas P ε is a closed and symmetric subset of H. Moreover, H \ P ε contains only sign changing functions.

Note that I ′ is a p.g.v.f for I, we can obtain a flow σ :   B → E satisfying (3.1) for all ( t ,   u ) ∈ B : = { ( t , u ) : u ∈ H ¯ ,   0 ≤ t < ω + ( u ) } , where ω + ( u ) ∈ ( 0,   + ∞ ] is the maximal existence time for the trajectory σ ( t ,   u ) . We call σ the descending flow associated with I ′ . A subset M ⊂ H is invariant for the σ if

σ ( t , u ) ∈ M ,     forevery   u ∈ M   andevery     t ∈ [ 0, ω + ( u ) ) .

If M is an invariant subset of H, we also consider

A ( M ) : = { u ∈ H :   σ ( t , u ) ∈ M   forsome     t ∈ ( 0, ω + ( u ) ) } ,

A 0 : = { u ∈ H :   σ ( t , u ) → 0     as     t → ω + ( u ) } .

Note that A 0 is open.

4. Three Solutions with One Changing Sign

In this section, we will give some proposition for finding three solutions with one changing sign.

Proposition 4.1 Suppose W is a finite dimensional subspace of H, there holds:

1) sup J ( W ) < + ∞ ;

2) If sup J ( C ) < + ∞ , where C : = { t u :   t ≥ 0,   u ∈ S } and S is a closed subset of some finite dimensional subspace W of H, then there is a constant R > 0 such that

J ( u ) < − 1 2 ‖ u ‖ 2     for     u ∈ C \ B R ( 0 ) . (4.1)

Proof. 1) Obviously.

2) If we define

Φ ( u ) : = 1 2 ‖ u ‖ 2 − ∫ R N F ( x , u ( x ) ) d x       for     x ∈ R N ,   u ∈ H ,

then

J ( u ) = Φ ( u ) − 1 2 ‖ u ‖ 2     for     u ∈ H .

Inequality (2.3) implies that for any δ > 0 there exist constants a 1 ,   a 2 > 0 , such that

Φ ( u ) ≤ 1 2 ‖ u ‖ 2 − K ( δ ) ∫ R N | u | η d x + δ ∫ R N | u | 2 d x ≤ a 1 ‖ u ‖ 2 − a 2 ‖ u ‖ η .

Hence for R = ( a 1 a 2 ) 1 η − 2 we have

Φ ( u ) < 0 ,     for   u ∈ C \ B R ( 0 ) .

Thus (4.1) hold. □

Using (2.7), we can note that

I ′ ( u ) = u − K G ( u ) T ( u ) = u − A ( u )     for     u ∈ S r , (4.2)

where A ( u ) : = K G ( u ) T ( u ) .

Proposition 4.2 There exists ε 0 > 0 such that for 0 < ε ≤ ε 0 , there holds

1) If u ∈ P ε ± ∩ S r and t ∈ [ 0,   ω + ( u ) ) , then σ ( t ,   u ) ∈ i n t ( P ε ± ) ∩ S r ;

2) Every nontrivial solution u ∈ P ε − ∩ S r of (1.1) is negative, and every nontrivial solution u ∈ P ε + ∩ S r of (1.1) is positive.

Proof. 1) Let d : = 1 2 inf x ∈ R N V ( x ) > 0 , u ∈ S r and v = K G ( u ) , then

‖ u + ‖ L 2 = min y ∈ P − ‖ u − y ‖ L 2 ≤ 1 2 d min y ∈ P − ‖ u − y ‖ = 1 2 d d i s t ( u ,   P − ) .

Similarly, using (2.1) we find for every s ∈ [ 2,   2 * ] , there is a constant C s > 0 with

‖ u ± ‖ L s ≤ C s d i s t ( u ,   P ∓ ) . (4.3)

Since v − ∈ P − ,   v − v − = v + ∈ P + , we have

d i s t ( v ,   P − ) ≤ ‖ v − v − ‖ = ‖ v + ‖ .

It follows from (2.2), (2.9) and (4.3) that

d i s t ( v ,   P − ) ‖ v + ‖ ≤ ‖ v + ‖ 2 = 〈 v ,   v + 〉 = 〈 K G ( u ) ,   v + 〉 = ∫ R N     f ( x , u ) v + d x ≤ ∫ R N     f ( x , u ) + v + d x = ∫ R N     f ( x , u + ) v + d x ≤ ∫ R N ( d | u + | + K ( d ) | u + | p − 1 ) v + d x ≤ d ‖ u + ‖ L 2 ‖ v + ‖ L 2 + K ( d ) ‖ u + ‖ L p p − 1 ‖ v + ‖ L p ≤ ( 1 2 d i s t ( u ,   P − ) + K ¯ d i s t ( u ,   P − ) p − 1 ) ‖ v + ‖ ,

with a constant K ¯ > 0 . Hence

d i s t ( v ,   P − ) ≤ 1 2 d i s t ( u ,   P − ) + K ¯ d i s t ( u ,   P − ) p − 1 .

So there exists ε 0 > 0 such that

d i s t ( K G ( u ) ,   P − ) ≤ 3 4 d i s t ( u ,   P − )

for every u ∈ P ε − ∩ S r with 0 < ε ≤ ε 0 . Thus

K G ( u ) ∈ i n t ( P ε − )       for     u ∈ P ε − ∩ S r . (4.4)

For any u ∈ P ε − , we can choose δ > 0 small enough such that

u + h ( T ( u ) u + K G ( u ) ) = ( 1 + h T ( u ) ) u + h K G ( u ) ∈ P ε −       for     h ∈ ( 0 ,   δ )

Thus

lim h ↓ 0 d i s t ( u − h I ′ ( u ) ,   P ε − ) h = 0     for     u ∈ ∂ P ε − . (4.5)

It follows from Lemma 3.4 that if σ ( t ,   u ) is the solution of (3.1), then it will hold that σ ( t ,   u ) ∈ P ε − for all t ∈ [ 0,   ω + ( u ) ) . So we can obtain from (4.4) that

K G ( σ ( t ,   u ) ) ∈ i n t ( P ε − )         for     u ∈ P ε − ∩ S r ,     t > 0. (4.6)

Set α ( t ) : = − ∫ 0 t     T ( σ ( s ,   u ) ) d s , then α ′ ( t ) > 0 ,   α ( t ) > 0 , and α ( t ) is strictly increasing. Applying (4.6), we have

K G ( σ ( t ,   u ) ) α ′ ( t ) ∈ i n t ( P ε − )       for     u ∈ P ε − ∩ S r . (4.7)

If we define B ( t ) : = K G ( σ ( t ,   u ) ) α ′ ( t ) , and F t : = { B ( s ) :   0 ≤ s ≤ t } , then F t is a compact set of H. According to (4.7), F t ⊂ i n t ( P ε − ) and hence c o ¯ F t ⊂ i n t ( P ε − ) , where c o ¯ F t is the closed convex hull of F t in H. Note that

1 e α ( t ) − 1 ∫ 0 t     e α ( s ) K G ( σ ( s ,   u ) ) d s = 1 e α ( t ) − 1 ∫ 1 e α ( t ) K G ( σ ( α − 1 ( ln s ) ,   u ) ) α ′ ( α − 1 ( ln s ) ) d s = lim m → + ∞ 1 m ∑ i = 1 m     A ( α − 1 ( ln ( 1 + i m ( e α ( t ) − 1 ) ) ) ) ∈ c o ¯ F t ⊂ i n t ( P ε − ) .

From (3.2) we get

σ ( t ,   u ) = e − α ( t ) u + 1 − e − α ( t ) e α ( t ) − 1 ∫ 0 t     e α ( s ) K G ( σ ( s ,   u ) ) d s . (4.8)

Denote F : = { u ,   1 e α ( t ) − 1 ∫ 0 t     e α ( s ) K G ( σ ( s ,   u ) ) d s } , then F is also a compact set of H. Using by (4.6) and (4.8), we obtain that

σ ( t ,   u ) ∈ c o ¯ F ⊂ i n t ( P ε − ) .

Hence σ ( t , u ) ∈ i n t ( P ε − ) ∩ S r for u ∈ P ε − ∩ S r and t ∈ [ 0,   ω + ( u ) ) . And σ ( t , u ) ∈ i n t ( P ε + ) ∩ S r for u ∈ P ε + ∩ S r and t ∈ [ 0,   ω + ( u ) ) can be proved analogously.

(2) Put w = − K G ( u ) T ( u ) , it follows from (2.9) that

d i s t ( w ,   P − ) ‖ w + ‖ ≤ ‖ w + ‖ 2 = 〈 w ,   w + 〉 = 〈 − K G ( u ) T ( u ) ,   w + 〉 = r 2 ( ∫ R N     f ( x , u ) u d x ) − 1 ∫ R N   f ( x , u ) w + d x .

Any u ∈ S r , 1 r 2 ∫ R N     f ( x ,   u ) u d x > 0 . By (2.2) and (4.3), for 0 < ξ < d r 2 ∫ R N     f ( x ,   u ) u d x < K 1 with a constant K 1 > 0 , we get

∫ R N   f ( x , u ) w + d x ≤ ∫ R N     f ( x , u ) + w + d x = ∫ R N     f ( x , u + ) w + d x ≤ ∫ R N ( ξ | u + | + K ( ξ ) | u + | p − 1 ) w + d x ≤ ξ ‖ u + ‖ L 2 ‖ w + ‖ L 2 + K ( ξ ) ‖ u + ‖ L p p − 1 ‖ w + ‖ L p ≤ ( δ 2 d d i s t ( u ,   P − ) + K 2 d i s t ( u ,   P − ) p − 1 ) ‖ w + ‖ ,

with a constant K 2 > 0 . So

d i s t ( w ,   P − ) ‖ w + ‖ ≤ ( 1 2 d i s t ( u ,   P − ) + K ˜ d i s t ( u ,   P − ) p − 1 ) ‖ w + ‖ ,

with a constant K ˜ > 0 . Thus

d i s t ( w ,   P − ) ≤ 1 2 d i s t ( u ,   P − ) + K ˜ d i s t ( u ,   P − ) p − 1 .

Hence, for ε 0 > 0 small enough

d i s t ( − K G ( u ) T ( u ) ,   P − ) ≤ 3 4 d i s t ( u ,   P − )

for every u ∈ P ε − ∩ S r with 0 < ε < ε 0 . In particular we have A ( ∂ P ε − ) ⊂ P ε − . If moreover u ∈ P ε − satisfies A ( u ) = u , then u ∈ P − . If finally u ≠ 0 , we conclude u ( x ) < 0 for all x by the maximum principle . Hence, every nontrivial solution u ∈ P ε − ∩ S r of (1.1) is negative. Similarly, every nontrivial solution u ∈ P ε + ∩ S r of (1.1) is positive. □

In view of Proposition 4.2, the next proposition just follows from Liu and Sun 

Lemma 4.1  Let E be a Hilbert space. Assume that Φ ∈ C 1 ( E ,   R ) , Φ ′ ( u ) = u − A ( u ) for u ∈ E , D 1 ∩ D 2 ≠ ∅ , and A ( ∂ D i ) ⊂ D i   ( i = 1 , 2 ) . Then there is a p.g.v.f Q for Φ which enables D 1 and D 2 to be invariant sets of descending flow and ∂ D i ⊂ C H ( D i ) ( i = 1 , 2 ) .

Lemma 4.2  Let E be a Hilbert space. Suppose Φ ∈ C 1 ( E ,   R ) satisfies (PS) and Φ ′ ( u ) has the expression Φ ′ ( u ) = u − A ( u ) for u ∈ E . Assume that D 1 and D 2 are open convex subset of E with the properties that D 1 ∩ D 2 ≠ ∅ and A ( ∂ D i ) ⊂ D i   ( i = 1 , 2 ) . If there exists a path h : [ 0,1 ] → E such that

h ( 0 ) ∈ D 1 \ D 2 ,   h ( 1 ) ∈ D 2 \ D 1 ,

and

inf u ∈ D 1 ¯ ∩ D 2 ¯ Φ ( u ) > inf t ∈ [ 0,1 ] Φ ( h ( t ) ) .

Then Φ has at least four critical points, one in D 1 ∩ D 2 , one in D 1 \ D 2 ¯ , one in D 2 \ D 1 ¯ , and one in E \ ( D 1 ¯ ∪ D 2 ¯ ) .

Note that I ′ is a p.g.v.f of I such that P ε + and P ε − are invariant for the associated descending flow. Moreover

∂ P ε ± ⊂ A ( P ε ± )       for     0 < ε ≤ ε 0 . (4.9)

holds following from Proposition 4.2 and Lemma 4.1.

Proposition 4.3 If 0 < ε ≤ ε 0 , then P ε + ¯ ∩ P ε − ¯ ⊂ A 0 . In particular there holds

J ( u ) > − 1 2 ‖ u ‖ 2     for     u ∈ P ε + ¯ ∩ P ε − ¯ \ { 0 } . (4.10)

Proof. First, from (A2) and (A3) we have

∫ R N   F ( x , u ) d x ≤ 1 η ∫ R N   f ( x , u ) u d x ≤ C η ( ‖ u ‖ L 2 2 + ‖ u ‖ L p p ) .

Since by (3.3) we infer that

‖ u ‖ L s 2 = ‖ u + + u − ‖ L s 2 = ‖ u + ‖ L s 2 + ‖ u − ‖ L s 2 ≤ C s 2 d i s t ( u ,   P − ) 2 + C s 2 d i s t ( u ,   P + ) 2 ≤ 2 ε 2 C s 2 .

Hence, for u ∈ P ε + ∩ P ε −

J ( u ) = 1 2 ‖ u ‖ 2 − ∫ R N     F ( x , u ) d x − 1 2 ‖ u ‖ 2 ≥ 1 2 ‖ u ‖ 2 − C η ( ‖ u ‖ L 2 2 + ‖ u ‖ L p p ) − 1 2 ‖ u ‖ 2 ≥ 1 2 ‖ u ‖ 2 − C η ( 2 ε 2 C s 2 + ( 2 ε 2 C s 2 ) p 2 ) − 1 2 ‖ u ‖ 2 ≥ − 1 2 ‖ u ‖ 2 .

Next we recall that P ε + ∩ P ε − contains no critical points of I by Proposition 4.2. Thus by (4.9), (4.10) and the invariance of P ε + ∩ P ε − we find that

σ ( t , u ) → 0     as   t → ω + ( u )

for every u ∈ P ε + ¯ ∩ P ε − ¯ as claimed. □

As a consequence of the preceding discussion, the existence of three solutions with one changing sign follows from Lemma 4.2.

Proof of Theorem 1.1 Choose v ∈ S r , and set

C : = { t v + + s v − :   t ≥ 0,   s ≥ 0 } .

It follow from Proposition 1, there exists R > 0 such that

J ( u ) < − 1 2 ‖ u ‖ 2   for     u ∈ C \ B R ( 0 ) .

By the choose of r, we have r > R , now we define the path

h : [ 0,1 ] → S r ,     h ( t ) = t r ‖ v + ‖ v + + ( 1 − t ) r ‖ v − ‖ v − ,

then

h ( 0 ) = r ‖ v − ‖ v − ∈ C ∩ P ε − ∩ S r ,       h ( 1 ) = r ‖ v + ‖ v + ∈ C ∩ P ε + ∩ S r ,

hence

sup t ∈ [ 0 , 1 ] I ( h ( t ) ) = sup t ∈ [ 0 , 1 ] J ( h ( t ) ) < − r 2 2 .

Applying Proposition 4.3, we obtain

J ( u ) > − 1 2 ‖ u ‖ 2     for     u ∈ P ε + ¯ ∩ P ε − ¯ \ { 0 } .

Thus

inf u ∈ P ε + ¯ ∩ P ε − ¯ I ( u ) > − r 2 2 .

So we have

inf u ∈ P ε + ¯ ∩ P ε − ¯ I ( u ) > sup t ∈ [ 0 , 1 ] I ( h ( t ) ) .

Since I satisfies (PS), I ′ ( u ) = u − A ( u ) for u ∈ S r , P ε + and P ε − are open convex subsets of H, P ε + ∩ P ε − ∩ S r ≠ ∅ . And by Proposition 4.2, A ( P ε ± ¯ ∩ S r ) ⊂ P ε ± ¯ ∩ S r . It follows from Lemma 4.2 that I has a critical points u + ∈ ( P ε + \ P ε − ) ∩ S r , u − ∈ ( P ε − \ P ε + ) ∩ S r , u ¯ ∈ S r \ ( P ε + ¯ ∪ P ε − ¯ ) , where u + is a positive solution of (1.1), u − is a negative solution of (1.1) and u ¯ is a sign changing solution of (1.1). □

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