Nontrivial solutions for Schrödinger-Kirchhoff-type problem in R^N

In the present paper, we use variational methods to prove two existence results of nontrivial solutions for the Schrödinger-Kirchhoff-type problem

One deals with the asymptotic behaviors of f near zero and infinity and the other deals with 4-superlinearity of F at infinity.

In this paper, we consider the Schrödinger-Kirchhoff-type problem

where constants a>0, b\ge 0, N=1,2 or 3, V\in C(R^N,R) and f\in C(R^N\times R,R). We are concerned with the existence of nontrivial solutions of (1.1), corresponding to the critical points of the energy functional

where F(x,t)={\int }_0^{t}f(x,s)ds.

When a=1, b=0, problem (1.1) reduces to the following semilinear Schrödinger equation:

Equation (1.3) has been studied extensively by many authors, and there is a large body of literature on the existence and multiplicity of results of solutions for equation (1.3); for example, we refer the readers to [1–3] and references therein.

On the other hand, the Kirchhoff-type problem on a bounded domain \mathrm{\Omega }\subset R^N,

is related to the stationary analogue of the Kirchhoff equation

which was proposed by Kirchhoff [4] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. It is pointed out in [5] that Kirchhoff-type problem (1.4) models several physical and biological systems, where u describes the process which depends on the average of itself (for example, population density). For the case of bounded domain, some interesting studies by variational methods can be found in [6–14] for equation (1.4) with several growth conditions on f. Very recently, Kirchhoff-type equations on the unbounded domain or the whole space R^N have also attracted a lot of attention. Many solvability conditions on the nonlinearity have been given to obtain the existence and multiplicity of solutions for problem (1.1). In [15, 16], the authors studied the case of superlinear nonlinearity. In [17, 18], the authors considered the case of radial potentials. In [19], the authors studied the case of nonhomogeneous nonlinearity.

Equation (1.1) can be viewed as the combination of (1.3) and (1.4) in R^N. So we call it the Schrödinger-Kirchhoff-type problem. Compared with (1.3) or (1.4), problem (1.1) is much more complicated and R^N is in place of the bounded domain \mathrm{\Omega }\subset R^N in (1.4). This makes the study of equation (1.1) more difficult and interesting. In the present paper, the goal is to study the existence of nontrivial solutions for equation (1.1) for the case of asymptotical nonlinearity and weaker superlinear conditions compared with [15, 16].

For the potential V, we assume

1. (V)

   V\in C(R^N,R) satisfies infV(x)\ge \alpha >0 and for each M>0, , where α is a constant and meas denotes the Lebesgue measure in R^N.

Set

with the norm

Denote

with the inner product and the norm

Since infV(x)\ge \alpha >0, it is easy to see that the Hilbert space E is continuously embedded in H^1(R^N). By the Sobolev embedding theorem, we know the embedding

where 2^{\ast }=\frac{2N}{N-2}, if N\ge 3, and 2^{\ast }=+\mathrm{\infty }, if N=1,2, is also continuous, and there is a constant {\gamma }_s>0, 2\le s\le 2^{\ast }, such that

where

Moreover, the embedding E\hookrightarrow L^s(R^N) is compact for each due to the assumption (V). In fact, if N=3, it follows from Lemma 3.4 in [20]. If N=1,2, we also claim that the compactness of the embedding is valid for 2^{\ast }=+\mathrm{\infty }. Indeed, let \{u_n\}\subset E be a sequence of E such that u_n\rightharpoonup u weakly in E. Similarly to the proof of Lemma 3.4 in [20], up to a subsequence, we can obtain u_n\to u strongly in L^2(R^N). Next, we shall prove u_n\to u strongly in L^s(R^N) for . In fact, \mathrm{\forall }s\in (2,+\mathrm{\infty }), there are t\in (s,+\mathrm{\infty }) and \theta \in (0,1) such that s=2\theta +(1-\theta )t. Then, by the Hölder inequality,

Since the embedding

is continuous, \{u_n\} is bounded in L^t(R^N). This, together with {\parallel u_n-u\parallel }_2\to 0, shows u_n\to u strongly in L^s(R^N), , N=1,2. Therefore, the compactness result holds for N=1,2,3.

Throughout this paper, we shall always assume f(x,t)t\ge 0, \mathrm{\forall }(x,t)\in R^N\times R. To establish the existence of nontrivial solutions for Schrödinger-Kirchhoff-type problem (1.1) in R^N, we make the following assumptions:

(f₁) f\in C(R^N\times R,R) and |f(x,t)|\le c(1+{|t|}^{p-1}) for some , where c is a positive constant.

(f₂) f(x,t)=o(|t|) as |t|\to 0 uniformly in x\in R^N.

(f₃) {lim}_{|t|\to +\mathrm{\infty }}\frac{f(x,t)}{t^3}=l uniformly in x\in R^N.

(f₄) 4F(x,t)\le f(x,t)t+\alpha t^2 for all (x,t)\in R^N\times R, where ({\gamma }_2 appears in (1.5)).

We consider the subcritical case in the present paper, (f₂) and (f₃) with characterize the asymptotic behavior of f at zero and infinity. The condition (f₃) with l=+\mathrm{\infty } implies {lim}_{|t|\to +\mathrm{\infty }}\frac{F(x,t)}{t^4}=+\mathrm{\infty }, that is, 4-superlinearity of F at infinity.

As usual, the Ambrosetti-Rabinowitz [21] type condition

is assumed to ensure the boundedness of a Palais-Smale sequence. It implies that there exists a constant c>0 such that F(x,t)\ge c({|t|}^{\nu }-1). By a simple calculation, it is easy to see that (AR) implies that \underset{|t|\to +\mathrm{\infty }}{lim}\frac{F(x,t)}{t^4}=+\mathrm{\infty }, and hence (f₃) with does not ensure the condition (AR). Furthermore, our condition (f₃) with l\equiv +\mathrm{\infty } is much weaker than the condition (AR). It is important to show that there are many functions satisfying the conditions (f₁)-(f₄) with l\in (b{\mathrm{\Lambda }}^2,+\mathrm{\infty }) or l=+\mathrm{\infty }, where

But the condition (AR) is not satisfied.

Example 1 For any given M\in (b{\mathrm{\Lambda }}^2,+\mathrm{\infty }), set

Then it is easy to verify that f(x,t) satisfies (f₁)-(f₄) with l\equiv M, but does not satisfy the condition (AR).

Example 2 Set

Then it is easy to verify that f(x,t) satisfies (f₁)-(f₄) with l=+\mathrm{\infty }, but does not satisfy the condition (AR).

Our main results are stated as the following theorems.

Theorem 1.1 Let conditions (V) and (f₁)-(f₄) hold and l\in (b{\mathrm{\Lambda }}^2,+\mathrm{\infty }). Then problem (1.1) has at least one nontrivial solution in E.

Theorem 1.2 Let conditions (V), (f₁)-(f₄) with l\equiv +\mathrm{\infty } hold. Then problem (1.1) has at least one nontrivial solution in E.

Corollary 1.3 If the following (f₅) or (f₆) is used in place of (f₄):

(f₅) 4F(x,t)\le f(x,t)t, \mathrm{\forall }x\in R^N, \mathrm{\forall }t\in R.

(f₆) \frac{f(x,t)}{{|t|}^3} is nondecreasing with respect to t.

Then the conclusions of Theorem 1.1 and Theorem 1.2 hold.

First, under assumptions (V), the embedding E\hookrightarrow L^s(R^N) is compact for each . Then the condition (f₁) implies I\in C^1(E,R),

for all u,v\in E, and the weak solutions of problem (1.1) correspond to the critical points of energy functional I.

Recall that we say that I satisfies the (\mathit{PS}) condition at the level c\in R ({(\mathit{PS})}_c condition for short) if any sequence \{u_n\}\subset E along with I(u_n)\to c and I^{\prime }(u_n)\to 0 as n\to \mathrm{\infty } possesses a convergent subsequence. If I satisfies the {(\mathit{PS})}_c condition for each c\in R, then we say that I satisfies the (\mathit{PS}) condition.

For the proof of our main results, we will make use of the following lemmas.

Lemma 2.1 \mathrm{\Lambda }>0 defined by (1.6) achieves by some {\phi }_{\mathrm{\Lambda }}\in E with {\int }_{R^N}{\phi }_{\mathrm{\Lambda }}^{4}dx=1 and {\phi }_{\mathrm{\Lambda }}>0 a.e. in R^N.

Proof By the Sobolev embedding theorem, one has \mathrm{\Lambda }>0. In order to prove that the infimum is achieved, we consider a minimizing sequence \{u_n\}\subset E such that

By the embedding E\hookrightarrow L^s(R^N) is compact for each , up to a subsequence, we may assume that there is {\phi }_{\mathrm{\Lambda }}\in E such that

so that

because of the weak lower semicontinuity of {\parallel \cdot \parallel }_E^2. Furthermore, we may assume that {\phi }_{\mathrm{\Lambda }}(x)>0 a.e. in R^N. Otherwise, we can replace {\phi }_{\mathrm{\Lambda }} by |{\phi }_{\mathrm{\Lambda }}|. □

Lemma 2.2 Set Q(u)={\int }_{R^N}{|\mathrm{\nabla }u|}^{2}dx, u\in E. Then Q is weakly lower semicontinuous on E.

Proof The proof has been given by Lemma 2 in [16]. Next we give another direct method to prove it, which is much easier than Lemma 2 in [16]. Let \{u_n\}\subset E and u_n\rightharpoonup u in E. By the embedding E\hookrightarrow L^2(R^N) is compact and the weak lower semicontinuity of {\parallel \cdot \parallel }_E^2, then

This shows that Q is weakly lower semicontinuous on E. □

Lemma 2.3 Let (f₄) hold. Then any (\mathit{PS}) sequence of I is bounded in E.

Proof Let \{u_n\}\subset E be a (\mathit{PS}) sequence with

Hence, for large n, the combination of (1.5) with (f₄) implies that

Therefore, the conclusion follows from (2.2) and . □

Proof of Theorem 1.1 To begin with, we prove that there exist \rho ,\beta >0 such that I(u)\ge \beta for all u\in E with {\parallel u\parallel }_E=\rho, and I(t{\phi }_{\mathrm{\Lambda }})\to -\mathrm{\infty } as t\to +\mathrm{\infty }. Indeed, for any ϵ>0, by (f₁), (f₂) and (f₃), there exists C(\epsilon )>0 such that

Choosing ({\gamma }_2 appears in (1.5)), by (1.5) and (3.1),

Therefore, we can choose small \rho >0 such that

whenever u\in E with {\parallel u\parallel }_E=\rho.

Since l>b{\mathrm{\Lambda }}^2, by Fatou’s lemma and (f₃), we have

Hence, I(t{\phi }_{\mathrm{\Lambda }})\to -\mathrm{\infty } as t\to +\mathrm{\infty }. Therefore, we can find large t_0>0 such that {\parallel t_0{\phi }_{\mathrm{\Lambda }}\parallel }_E>\rho, .

Now, we prove that I satisfies the (\mathit{PS}) condition. Indeed, if a sequence \{u_n\}\subset E is such that

then by Lemma 2.3, \{u_n\} is bounded in E. Since the embedding E\hookrightarrow L^s(R^N) is compact for each s\in [2,2^{\ast }), up to a subsequence, there is u\in E such that

By (2.1) and a simple computation, we conclude

Then (3.2) implies that

Define the functional h_u: E\to R by

Obviously, h_u is a linear functional on E. Furthermore,

which implies that h_u is bounded on E, where Q(u) is defined in Lemma 2.2. Hence h_u\in E^{\ast }. Since u_n\rightharpoonup u in E, it has {lim}_{n\to \mathrm{\infty }}{h}_u(u_n)=h_u(u), that is, {\int }_{R^N}\mathrm{\nabla }u\cdot \mathrm{\nabla }(u_n-u)dx\to 0 as n\to \mathrm{\infty }. Consequently, by the boundedness of \{u_n\}, it has

Moreover, for any \epsilon >0, by (f₁), (f₂) and (f₃), there exists C(\epsilon )>0 such that

Hence, (3.5) implies

where c_1 and c_2 are independent of ε and n. Since {\parallel u_n-u\parallel }_4\to 0 as n\to +\mathrm{\infty }, we conclude

Since I^{\prime }(u_n)\to 0, the combination of (3.3), (3.4) and (3.6) implies that {\parallel u_n-u\parallel }_E\to 0.

Note that I(0)=0 applying the mountain pass theorem (Theorem 2.2 in [21]), then I possesses a critical value c\ge \beta, i.e., problem (1.1) has a nontrivial solution in E. This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2 Set ({\gamma }_2 appears in (1.5)). By (f₁) and (f₂), there exists C(\epsilon )>0 such that

Then, by (3.7), one has

Since p>2, there exist constants \rho ,\beta >0 such that I(u)\ge \beta for all u\in E with {\parallel u\parallel }_E=\rho (see the proof of Theorem 1.1).

Since E\hookrightarrow L^2(R^N) and L^2(R^N) is a separable Hilbert space, E has a countable orthogonal basis \{e_i\}. Set E_k=span\{e_1,\dots ,e_k\} and Z_k=E_k^{\mathrm{\perp }}. Since all norms are equivalent in a finite dimensional space, there is a constant C_4>0 such that

By (f₁)-(f₃) with l\equiv +\mathrm{\infty }, for any M>\frac{b}{4C_4^4}, there is a constant C(M)>0 such that

Hence, combining (3.8) and (3.9), we deduce

for all u\in E_k. Consequently, there is a point e\in E with {\parallel e\parallel }_E>\rho such that .

Next, we show that I satisfies the (\mathit{PS}) condition. In fact, let \{u_n\}\subset E be a (\mathit{PS}) sequence with

Then, by Lemma 2.3, \{u_n\} is bounded in E. Since the embedding E\hookrightarrow L^s(R^N) is compact for each s\in [2,2^{\ast }), up to a subsequence, there is u\in E such that

Thanks to (3.7), we have

where c_1 and c_2 are independent of ε and n. Since {\parallel u_n-u\parallel }_p\to 0 as n\to +\mathrm{\infty }, we conclude

The rest of the proof is the same as the proof of Theorem 1.1, the desired conclusion follows from the mountain pass theorem (Theorem 2.2 in [21]). This completes the proof of Theorem 1.2. □

Proof of Corollary 1.3 In order to obtain the desired conclusions, by the proof of Theorem 1.1 and Theorem 1.2, it is sufficient to show that the condition (f₅) or (f₆) implies the condition (f₄). First, it is obvious that (f₅) implies (f₄). Next, we only show (f₆) implies (f₄). Indeed, by (f₆), whenever u>0,

Whenever ,

Thus, (3.10) and (3.11) imply that 4F(x,u)\le f(x,u)u, \mathrm{\forall }x\in R^N, \mathrm{\forall }u\in R. Hence (f₆) implies (f₄). □

Remark In [16], Wu considered the superlinear nonlinearity case and obtained the existence of nontrivial solutions for problem (1.1) under the same conditions of our Corollary 1.3 with superlinear case (see Theorem 1 and Theorem 2 in [16]). Therefore, our Theorem 1.2 is a generalization of Theorem 1 and Theorem 2 in [16].

The author would like to thank the referees for several comments and valuable suggestions. This work is partly supported by the National Natural Science Foundation of China (11361048), the Foundation of Education of Commission of Yunnan Province (2012Y410, 2013Y015) and the Youth Program of Yunnan Provincial Science and Technology Department (2013FD046).

Affiliations

1. College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan, 655011, P.R. China

   • Bitao Cheng

Corresponding author

Correspondence to Bitao Cheng.

Competing interests

The author declares that he has no competing interests.

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