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Solutions of perturbed p-Laplacian equation with critical nonlinearity and magnetic fields

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Abstract

In this paper, we consider a perturbed p-Laplacian equation with criticalnonlinearity and magnetic fields on R N . By using the variational method, we establish theexistence of nontrivial solutions of the least energy.

MSC: 35B33, 35J60, 35J65.

1 Introduction

In this paper, we are concerned with the existence of nontrivial solutions to thefollowing perturbed p-Laplacian equation with critical nonlinearity and magnetic fields of the form

‒ε p Δ p , A u+V(x) | u | p 2 u=K(x) | u | p * 2 u+f ( x , | u | p ) | u | p 2 u,xϵ R N ,
(1.1)

where Δ p , A u=div( | Δ u + i A ( x ) u | p 2 (u+iA(x)u)), i is the imaginary unit, A(x): R N R N is a real vector potential, 1<p<N, p * =Np/(Np) denotes the Sobolev critical exponent andN3.

This paper is motivated by some works concerning the nonlinear Schrödinger equation withmagnetic fields of the form

i h δ ψ δ t = ħ 2 2 m ( Δ + i A ( x ) ) 2 ψ + W ( x ) ψ K ( x ) | ψ | 2 _ ψ 2 ψ - h ( x , | ψ | 2 ) ψ for x x ϵ R N ,
(1.2)

where ħ is Planck's constant, i is the imaginary unit, 2 * = 2 N N - 2 (N3) is the critical exponent, A(x): R N R N is a real vector potential, B=curlA and W(x) is a scalar electric potential.

In physics, we are interested in the standing wave solutions, that is, solutions to(1.2) of the type

ψ(x,t)=exp ( - i E t ħ ) u(x),

where ħ is a sufficiently small constant, E is a real number, andu(x) is a complex-valued function satisfying

- ( + i A ( x ) ) 2 u(x)+λ ( W ( x ) - E ) u(x)=λK(x) | u | 2 * - 2 u+λh ( x , | u | 2 ) u,xϵ R N .
(1.3)

We can conduct the transition from quantum mechanics to classical mechanics by lettingħ0. Thus, the existence of semiclassical solutions has agreat charm in physical interest.

Problem (1.3) with A(x)0 has an extensive literature. Different approaches havebeen taken to investigate this problem under various hypotheses on the potential andnonlinearity. See for example [118] and the references therein. The above-mentioned papers mostly concentrated onthe nonlinearities with subcritical conditions. Floer and Weinstein in [11] first studied the existence of single and multiple spike solutions based onthe Lyapunov-Schmidt reductions. Subsequently, Oh [16, 17] extended the results in a higher dimension. Kang and Wei [14] established the existence of positive solutions with any prescribed number ofspikes, clustering around a given local maximum point of the potential function. Inaccordance with the Sobolev critical nonlinearities, there have been many papers devotedto studying the existence of solutions to elliptic boundary-valued problems on boundeddomains after the pioneering work by Brézis and Nirenberg [4]. Ding and Lin [8] first studied the existence of semi-classical solutions to the problem on thewhole space with critical nonlinearities and established the existence of positivesolutions, as well as of those that change sign exactly once. They also obtainedmultiplicity of solutions when the nonlinearity is odd.

As far as problem (1.3) in the case of A(x)=0 is concerned, we recall Bartsch [2], Cingolani [5] and Esteban and Lions [10]. This kind of paper first appeared in [10]. The authors obtained the existence results of standing wave solutions forfixed ħ>0 and special classes of magnetic fields. Cingolani [5] proved that the magnetic potential A(x) only contributes to the phase factor of the solitarysolutions for ħ>0 sufficiently small. For more results, we refer the readerto [1921] and the references therein.

For general p1, most of the works studied the existence results toequation (1.1) with A(x)0. See, for example, [2228] and the references therein. These papers are mostly devoted to the study ofthe existence of solutions to the problem on bounded domains with the Sobolevsubcritical nonlinearities.

However, to our best knowledge, it seems that there is no work on the existence ofsemiclassical solutions to perturbed p-Laplacian equation on R N involving critical nonlinearity and magnetic fields. Inthis paper, we consider problem (1.1) with magnetic fields. The main difficulty in thecase is the lack of compactness of the energy functional associated to equation (1.1)because of unbounded domain R N and critical nonlinearity. At the same time, we mustconsider complex-valued functions for the appearance of electromagnetic potentialA(x). To overcome this difficulty, we chiefly follow the ideasof [5]. Notice that although the ideas were used in other problems, the adaption ofthe procedure to our problem is not trivial at all. We need to make careful and complexestimates and prove that the energy functional possesses a Palais-Smale sequence, whichhas a strongly convergent sequence.

We make the following assumptions on V(x), A(x), f(x,s) and K(x) throughout the paper:

(V0) VϵC( R N ), V(0)= inf x ϵ R N V(x)=0, and there exists b>0 such that the set ν b :={xϵ R N :V(x)<b} has a finite Lebesgue measure;

(A0) AC( R N , R N ) and A(0)=0;

(K0) K(x)C( R N , R + ), 0<infKsupK<;

(H1) fC( R N × R + ,R) and f(x,s)=o(|s|) uniformly in x as s0;

(H2) there are c 1 >0 and p<α< p * such that |f(x,s)| c 1 (1+ | s | α - p p ) for all (x,s);

(H3) there exist a 0 >0, q>p and θ(p, p * ) such that F(x,s)p a 0 | s | q p and θ p F(x,s)f(x,s)s for all (x,s), where F(x,s)= ʃ 0 s f(x,t)dt.

Our main result is the following.

Theorem 1 Assume that (V0), (A0), (K0)and (H1)-(H3) hold. Then foranyσ>0, there exists ε >σsuch that ifε ε σ , equation (1.1) has at least one positiveleast energy solution u ε , which satisfies

θ p p θ ʃ R N ( ε p | u ε | p + V ( x ) | u ε | p ) σ ε N .

The paper is organized as follows. In Section 2, we give some necessary preliminaries.Section 3 is devoted to the technical lemmas. The proof of Theorem 2 is given in thelast section.

2 Preliminaries

Let λ= ε - p . Equation (1.1) reads then as

- Δ p , A u+λV(x) | u | p - 2 u=λK(x) | u | p * - 2 u+λf ( x , | u | p ) | u | p - 2 u,xϵ R N .
(2.1)

We are going to prove the following result.

Theorem 2 Assume that (V0), (A0), (K0)and (H1)-(H3) are satisfied. Then foranyσ>0, there exists λ σ >0such that ifλ> λ σ , then equation (2.1) has at least onesolution of least energy u ε satisfying

θ - p p θ ʃ R N ( | u λ | p + λ V ( x ) | u λ | p ) σ λ 1 - N p .
(2.2)

In order to prove these theorems, we introduce the space

E λ , A = { u ϵ W 1 , p ( R N , C ) : ʃ R N λ V ( x ) | u | p < , λ > 0 } ,

equipped with the norm

|| u || λ , A = ( ʃ R N ( | u + i λ 1 p A ( x ) u | p + λ V ( x ) | u | p ) ) 1 p .

It is known that E λ , A is the closure of C 0 ( R N ,C). Similar to the diamagnetic inequality [10], we have the following inequality

| | u ( x ) | | | u + i λ 1 p A u | .

In fact, since A(x) is real-valued, one has

| | u ( x ) | | | u u ¯u | u | | = | Re ( u + i λ 1 p A u ) u u | u | | | u + i λ 1 p A u |
(2.3)

(the bar denotes a complex conjugation). This inequality implies that ifu E λ , A , then |u| W 1 , p ( R N ), and, therefore, u L q ( R N ) for any q[p, p * ). That is, if u n u in E λ , A , then u n u in L loc q ( R N ) for any qϵ[p, p * ) and u n u a.e. in R N .

Solutions of (2.1) will be sought in the Sobolev space E λ , A as critical points of the functional

I λ ( u ) = 1 p ʃ R N ( | u + i λ 1 p A u | p + λ V ( x ) | u | p ) - λ p * ʃ R N K ( x ) | u | p * - λ p ʃ R N F ( x , | u | p ) = 1 p ǁ u ǁ λ , A - λ ʃ N R N G ( x , u ) ,

where G(x,u)= 1 p * K(x) | u | p * + 1 p F(x, | u | p ).

It is easy to see that I λ is a C 1 -functional on E λ , A [29].

3 Behavior of ( PS ) c sequence and a mountain pass structure

In this section, we commence by establishing the necessary results which complete theproof of Theorem 2.

Lemma 3.1 Let (V0), (A0), (K0)and (H1)-(H3) be satisfied. Forthe ( PS ) c sequence{ u n } E λ , A for I λ , we get thatc0and{ u n }is bounded in the space E λ , A .

Proof Under assumptions (K0) and (H3), we have

I λ ( u n ) - 1 θ I ' λ ( u n ) u n = ( 1 p - 1 θ ) ǁ u n ǁ p λ A p + ( 1 θ - 1 p * ) λ ʃ R N K ( x ) | u n | p * + λ ʃ R N ( 1 θ f ( x , | u n | p ) | u n | p - 1 p F ( x , | u n | p ) ) .

In connection with the facts that I λ ( u n )c and I ' λ ( u n )0 as n, we obtain that the ( PS ) c sequence { u n } is bounded in E λ , A , and the energy level c0.

Next, let { u n } denote a ( PS ) c sequence. By Lemma 3.1, it is bounded, thus, without lossof generality, we may assume that u n u in E λ , A . Furthermore, passing to a subsequence, we have u n u in L loc q ( R N ) for any qϵ[p, p * ) and u n u a.e. in R N .

Lemma 3.2 For anysϵ[p, p * ), there is a subsequence{ u n j }such that for anyε>0, there exists r ε >0with

lim j sup ʃ B j \ B r | u n j | s εfor anyr r ε ,

where B r :={xϵ R N :|x|r}.

Proof It is easily obtained by the similar proof of Lemma 3.2 [8].

Let η C ( R + ) be a smooth function satisfying 0η(t)1, η(t)=1 if t1 and η(t)=0 if t2. Define u ~ j (x)=η(2|x|/j)u(x). It is not difficult to see that

ǁ u ǁ u ~ j ǁ λ , A 0asj.

Lemma 3.3 One has

lim j sup Re ʃ R N ( f ( x , | u n j | p ) | u n j | p - 2 u n j - f ( x , u n j - u ~ j ) | u n j - u ~ j | p - 2 ( u n j - u ~ j ) - f ( x , | u ~ j | p ) | u ~ j | p - 2 u ~ j ) ϕ = 0

uniformly inϕ E λ , A with ǁ ϕ ǁ λ , A 1.

Proof By direct computation, we easily obtain u ~ j u in E λ , A . The local compactness of the Sobolev embedding impliesthat, for any r0, we have

lim j sup Re ʃ B r ( f ( x , | u n j | p ) | u n j | p - 2 u n j - f ( x , u n j - u ~ j ) | u n j - u ~ j | p - 2 ( u n j - u ~ j ) - f ( x , | u ~ j | p ) | u ~ j | p - 2 u ~ j ) - ϕ = 0

uniformly in ǁ ϕ ǁ λ , A 1. For any ε>0, there is r ε 0 such that

lim j sup ʃ B j \ B r | u ~ j | s ʃ R N \ B r | u | s ε

for all r r ε . By the assumptions and the Hölder inequality, we have

lim j sup Re ʃ R N ( f ( x , | u n j | p ) | u n j | p - 2 u n j - f ( x , u n j - u ~ j ) | u n j - u ~ j | p - 2 ( u n j - u ~ j ) - f ( x , | u ~ j | p ) | u ~ j | p - 2 u ~ j ) ϕ = lim j sup Re ʃ B j \ B r ( f ( x , | u n j | p ) | u n j | p - 2 u n j - f ( x , u n j - u ~ j ) | u n j - u ~ j | p - 2 ( u n j - u ~ j ) - f ( x , | u ~ j | p ) | u ~ j | p - 2 u ~ j ) ϕ c 1 lim j sup ʃ B j \ B r ( | u n j | p - 1 + | u ~ j | p - 1 ) | ϕ | + c 2 lim j sup ʃ B j \ B r ( | u n j | ǁ ǁ 1 + | u ~ j | ǁ ǁ 1 ) | ϕ | c 1 lim j sup ( ǁ u n j ǁ L p p - 1 ( B j \ B r ) + ǁ u ~ j ǁ L p p - 1 ( B j \ B r ) ) ǁ ϕ ǁ L p ( B j \ B r ) + c 2 lim j sup ( ǁ u n j ǁ L α ( B j \ B r ) α - 1 + ǁ u ~ j ǁ L α ( B j \ B r ) α - 1 ) ǁ ϕ ǁ L α ( B j \ B r ) c 3 ε p - 1 p + c 4 ε α - 1 α .

This proof is completed.

Lemma 3.4 One has along a subsequence

I λ ( u n - u ~ n )c- I λ (u)

and

I ' λ ( u n - u ~ n )0in E λ - 1 (the dual space of E λ ).

Proof Combining Lemma 2.1 of [30] and the arguments of [31], one has

I λ ( u n - u ~ n ) = I λ ( u n ) - I λ ( u ~ n ) + λ p * ʃ R N K ( x ) ( | u n | p * - | u n - u ~ n | p * - | u ~ n | p * ) + λ p * R N ( F ( x , | u n | p ) - F ( x , | u n - u ~ n | p ) - F ( x , | u ~ n | p ) ) + o ( 1 ) .

By the Brézis-Lieb lemma [32], we get

lim n ʃ R N K(x) ( | u n | p * - | u n - u ~ n | p * - | u ~ n | p * ) =0

and

lim n ʃ R N ( F ( x , | u n | p ) - F ( x , | u n - u ~ n | p ) - F ( x , | u ~ n | p ) ) =0.

We now observe that I λ ( u n )c and I λ ( u ~ n ) I λ (u), which gives

I λ ( u n - u ~ n )c- I λ (u).

Moreover, by direct computation, we get

I λ ' ( u n - u ~ n ) ϕ = I ' λ ( u n ) ϕ - I ' λ ( u ~ n ) ϕ + λ Re ʃ R N K ( x ) ( | u n | p * - 2 u n - | u n - u ~ n | p * - 2 ( u n - u ~ n ) - | u ~ n | p * - 2 u ~ n ) ϕ + λ Re ʃ R N ( f ( x , | u n | p ) | u n | p - 2 u n - f ( x , | u n - u ~ n | p ) | u n - u ~ n | p - 2 ( u n - u ~ n ) - f ( x , | u ~ n | p ) | u ~ n | p - 2 u ~ n ) ϕ + o ( 1 ) .

It then follows from the standard arguments that

lim n Re ʃ R N K(x) ( | u n | p * - 2 u n - | u n - u ~ n | p * - 2 ( u n - u ~ n ) - | u ~ n | p * - 2 u ~ n ) ϕ=0

uniformly in ǁ ϕ ǁ λ , A 1. Combining Lemma 3.3, we get I ' λ ( u n - u ~ n )0. The proof is completed.

Let u n 1 = u n - u ~ n , then u n -u= u n 1 +( u ~ n -u). Therefore, u n u in E λ , A if and only if u n 1 0 in E λ , A .

Note that

I λ ( u n 1 ) - 1 p I λ ( u n 1 ) u n 1 = ( 1 p - 1 p * ) λ ʃ R N K ( x ) | u n 1 | p * + λ ʃ R N 1 p ( | u n 1 | p f ( x , | u n 1 | p ) - F ( x , | u n 1 | p ) ) λ N ʃ R N K ( x ) | u n 1 | p * λ N K min ǁ u n 1 ǁ p * p * ,

where K min = inf x ϵ R N K(x)>0. Together with Lemma 3.4, one has

ǁ u n 1 ǁ p * p * N ( c - I λ ( u ) ) λ K min +o(1).
(3.1)

In the following, we consider the energy level of the functional I λ below which the ( PS ) c condition holds.

Denote V b (x):=max{V(x),b}, where b is the positive constant in assumption(V0). Since the set ν b has a finite measure, combining the fact that u n 1 0 in L loc p ( R N ), we get

ʃ R N V(x) | u n 1 | p = ʃ R N V b (x) | u n 1 | p +o(1).
(3.2)

Furthermore, by (K0) and (H1)-(H3), there exists C b >0 such that

ʃ R N ( K ( x ) | u n 1 | p * + | u n 1 | p f ( x , | u n 1 | p ) ) b ǁ u n 1 ǁ p p + C b ǁ u n 1 ǁ p * p * .
(3.3)

Let S be the best Sobolev constant of the immersion

S ǁ u ǁ p * p ʃ R N | u | p for alluϵ W 1 , p ( R N ) .

Lemma 3.5 There exists α 0 >0 (independent of λ) such that,for any ( PS ) c sequence{ u n } E λ , A for I λ with u n u, either u n uin E λ , A orc- I λ (u) α 0 λ 1 - N p .

Proof Arguing by contradiction, assume that u n u, then

lim inf n ǁ u n 1 ǁ λ , A >0.

Combining the Sobolev inequality, (3.2) and (3.3), we get

S ǁ u n 1 ǁ p * p ʃ R N | u n 1 | p ʃ R N ( | u n 1 + i λ 1 p A ( x ) u n 1 | p + λ V ( x ) | u n 1 | p ) - λ ʃ R N V ( x ) | u n 1 | p = λ ʃ R N ( K ( x ) | u n 1 | p * + | u n 1 | p f ( x , | u n 1 | p ) ) - λ ʃ R N V b ( x ) | u n 1 | p + o ( 1 ) λ b ǁ u n 1 ǁ p p + λ C b ǁ u n 1 ǁ p * p * - λ b ǁ u n 1 ǁ p p + o ( 1 ) = λ C b ǁ u n 1 ǁ p * p * + o ( 1 ) ,

which further gives

S λ C b ǁ u n 1 ǁ p * p * - p + o ( 1 ) λ C b ( N ( c - I λ ( u ) ) λ K min ) p N + o ( 1 ) = λ 1 - p N C b ( N K min ) p N ( c - I λ ( u ) ) p N + o ( 1 ) .

Denote α 0 = S N p C b - N p N 1 K min , then

α 0 λ 1 - N p c- I λ (u)+o(1).

We obtain the desired conclusion.

Lemma 3.6 There exists a constant α 0 >0 (independent of λ) such that ifa ( PS ) c sequence{ u n } E λ , A for I λ satisfiesc α 0 λ 1 - N p , the sequence{ u n }has a strongly convergent subsequencein E λ , A .

Proof By the fact that I λ (u)0 and Lemma 3.5, we easily get the requiredconclusion.

Now, we consider λ1 and prove that the energy functional I λ possesses the mountain pass structure.

Lemma 3.7 Under the assumptions of Theorem 2, thereexist α λ , ρ λ >0such that

I λ (u)>0if0< ǁ u ǁ λ , A < ρ λ and I λ (u) α λ if ǁ u ǁ λ , A = ρ λ .

Proof The proof of Lemma 3.7 is similar to the one of Lemma 4.1 in [8].

Lemma 3.8 For any finite dimensional subspaceF E λ , A , we have

I λ (u)-,uϵFas ǁ u ǁ λ , A .

Proof By assumptions (K0) and (H3), one has

I λ (u) 1 p ǁ u ǁ λ , A p -λ a 0 ǁ u ǁ q q for alluϵ E λ , A .

Since all norms in a finite-dimensional space are equivalent, in connection withq>p, we obtain the desired conclusion.

For λ large enough and c λ small sufficiently, I λ satisfies ( PS ) c λ condition by Lemma 3.6. Furthermore, we will find specialfinite-dimensional subspace, by which we establish sufficiently small minimaxlevels.

Define the functional

Ф λ (u)= 1 p ʃ R N ( | u + i λ 1 p A ( x ) u | p + λ V ( x ) | u | p ) -λ a 0 ʃ R N | u | q .

It is easy to see that Ф λ ϵ C 1 ( E λ , A ) and I λ (u) Ф λ (u) for all uϵ E λ , A . Note that

inf { ʃ R N | φ | p : φ ϵ C 0 ( R N , R ) , ǁ φ ǁ L q ( R N ) = 1 } =0.

For any δ>0, there is φ δ ϵ C 0 ( R N ,R) with ǁ φ δ ǁ L q ( R N ) =1 and supp φ δ B r δ (0) such that ǁ φ δ ǁ p p <δ. Let e λ (x)= φ δ ( λ p x), then supp e λ B λ - 1 p r δ (0). For any t0, we have

Ф λ ( t e λ ) = t p p ǁ e λ ǁ λ , A p - a 0 λ t q ʃ R N | φ δ ( λ p x ) | q = λ 1 - N p J λ ( t φ δ ) ,

where

J λ (u)= 1 p ʃ R N ( | u + i λ 1 p A ( x ) u | p + V ( λ - 1 p x ) | u | p ) a 0 ʃ R N | u | q .

We derive that

max t 0 J λ (t φ δ ) q - p p q ( q a 0 ) p q - p ( ʃ R N ( | φ δ | p + ( A ( λ - 1 p x ) + V ( λ - 1 p x ) ) | φ δ | p ) ) q q - p .

Observe that A(0)=0, V(0)=0 and ǁ φ δ ǁ p p <δ. Therefore, there exists Ʌ δ >0 such that for all λ Ʌ δ , we have

max t 0 I λ (t φ δ ) ( q - p p q ( q a 0 ) p q - p ( 5 δ ) q q - p ) λ 1 - N p .
(3.4)

Lemma 3.9 Under the assumptions of Theorem 2, foranyσ>0, there is Ʌ σ >0such that for eachλ Ʌ σ , there exists e - λ ϵ E λ , A with ǁ e - λ ǁ λ , A > ρ λ , I λ ( e - λ )0and

max t 0 I λ (t e ¯ λ )σ λ 1 - N p ,

where ρ λ is defined in Lemma 3.7.

Proof For any σ>0, we can choose δ<0 so small that

q - p p q ( q a 0 ) p q - p ( 5 δ ) q q - p σ.

Denote e λ (x)= φ δ ( λ p x) and Ʌ σ = Ʌ δ . Let t ¯ λ >0 be such that t ¯ λ ǁ e λ ǁ λ , A > ρ λ and I λ (t e λ )0 for all t t ¯ λ . Then, combining (3.4), e ¯ λ = t ¯ λ e λ meets the requirements.

4 Proof of Theorem 2

In this section, we give the proof of Theorem 2.

Proof By Lemma 3.9, for any σ>0 with 0<σ< α 0 , we choose Ʌ σ >0 and define the minimax value

c λ = inf γ ϵ Г λ max t ϵ [ 0 , 1 ] I λ ( γ ( t ) ) with c λ σ λ 1 - N p for eachλ Ʌ σ ,

where Г λ ={γϵC([0,1], E λ , A ):γ(0)=0,γ(1)= e - λ }.

Lemma 3.6 shows that I λ satisfies ( PS ) c λ condition. Therefore, by the mountain pass theorem, thereexists u λ ϵ E λ , A , which satisfies I λ ( u λ )= c λ and I λ ( u λ )=0. That is, u λ is a weak solution of (2.1). Furthermore, it is well knownthat u λ is the least energy solution of equation (2.1).

Moreover, together with I λ ( u λ )σ λ 1 - N p and I λ ( u λ )=0, we have

I λ ( u λ ) = I λ ( u λ ) - 1 θ I λ ( u λ ) ( u λ ) = ( 1 p - 1 θ ) ǁ u λ ǁ λ , A p + ( 1 θ - 1 p * ) λ ʃ R N K ( x ) | u λ | p * + λ ʃ R N ( 1 θ | u λ | p f ( x , | u λ | p ) - 1 p F ( x , | u λ | p ) ) ( 1 p - 1 θ ) ǁ u λ ǁ λ , A p .

By inequality (2.3), we obtain

θ - p p θ ʃ R N ( | u λ | p + λ V ( x ) | u λ | p ) σ λ 1 - N p .

The proof is complete.

Authors' contributions

The authors contributed equally in this article. They read and approved the finalmanuscript.

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Acknowledgements

The authors would like to appreciate the referees for their precious comments andsuggestions about the original manuscript. This research was supported by theNational Natural Science Foundation of China (11271364) and the Fundamental ResearchFunds for the Central Universities (2012QNA46).

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Correspondence to Huixing Zhang.

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Keywords

  • p-Laplacian equation
  • critical nonlinearity
  • magnetic fields
  • mountain pass theorem