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

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

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.

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].

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 $L^s(R^N)$, $2\le s<+\mathrm{\infty }$, $N=1,2$. 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 $2\le p<2^{\ast }$, 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 $0<\alpha <\frac{min\{a,1\}}{{\gamma }_2^2}$ (${\gamma }_2$ appears in (1.5)).

We consider the subcritical case in the present paper, (f₂) and (f₃) with $l<+\mathrm{\infty }$ 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.

But the condition (AR) is not satisfied.

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

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.

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 $R^N$.

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.

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.

Therefore, the conclusion follows from (2.2) and $\alpha {\gamma }_2^2<min\{a,1\}$. □

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

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

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. □

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).

for all $u\in E_k$. Consequently, there is a point $e\in E$ with ${\parallel e\parallel }_E>\rho$ such that $I(e)<0$.

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. □

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].