On the existence of solutions for nonhomogeneous Schrödinger-Poisson system

Many mathematicians have devoted their efforts to the study of (1.1) with various nonlinearities \(f(x,u)\). We recall some of them as follows.

The case of \(g\equiv0\), that is, the homogeneous case, has been studied widely in [6, 8, 11, 17–25]. In 2010, Cerami and Vaira [10] study system (1.1) in the case of \(f(x,u)=a(x)|u|^{p-2}u\) with \(4< p<6\) and \(a(x)>0\). In order to recover the compactness of the embedding of \(H^{1}(\mathbb{R}^{3})\) into the Lebesgue space \(L^{s}(\mathbb{R}^{3})\), \(s\in[2,6)\), they establish a global compactness lemma. They prove the existence of positive ground state and bound state solutions without requiring any symmetry property on \(a(x)\) and \(K(x)\).

In 2012, Sun et al. [26] consider a more general case, that is, \(f(x,u)=a(x)\tilde{f}(u)\) where f̃ is asymptotically linear at infinity, i.e. \(\tilde{f}(s)/s\rightarrow c\) as \(s\rightarrow+\infty\) with a suitable constant c. They establish a compactness lemma different from that in [10] and prove the existence of ground state solutions. In [27], Ye and Tang study the existence and multiplicity of solutions for homogeneous system of (1.1) when the potential V may change sign and the nonlinear term f is superlinear or sublinear in u as \(|u|\rightarrow\infty\). For the Schrödinger-Poisson system with sign-changing potential, see [14, 28].

Huang et al. [12] study the case that \(f(x,u)\) is a combination of a superlinear term and a linear term. More precisely, \(f(x,u)=k_{1}(x)|u|^{p-2}u+\mu h_{1}(x)u\), where \(4< p<6\) and \(\mu>0\), \(k_{1}\in C(\mathbb{R}^{3})\), \(k_{1}\) changes sign in \(\mathbb{R}^{3}\) and \(\lim_{|x|\rightarrow+\infty}k_{1}(x)=k_{\infty}<0\). They prove the existence of at least two positive solutions in the case that \(\mu>\mu_{1}\) and near \(\mu_{1}\), where \(\mu_{1}\) is the first eigenvalue of \(-\Delta+\mathrm{id}\) in \(H^{1}(\mathbb{R}^{3})\) and with weight function \(h_{1}\). In [29, 30], the authors consider the critical case of \(p=6\); in [31] one studies the case of \(p=4\).

Sun et al. [32] get infinitely many solutions for (1.1), where we have the nonlinearity \(f(x,u)=k_{2}(x)|u|^{q-2}u-h_{2}(x)|u|^{l-2}u\), \(1< q<2<l<\infty\), i.e. the nonlinearity involving a combination of concave and convex terms. For more results on the effect of concave and convex terms of elliptic equations, see [33, 34] and the references therein.

Next, we consider the nonhomogeneous case of (1.1), that is, \(g\not\equiv0\). The existence of radially symmetric solutions is obtained for above nonhomogeneous system in [23]. Chen and Tang [35] obtain two solutions for the nonhomogeneous system with \(f(x,u)\) satisfying Amborosetti-Rabinowitz type condition and V being non-radially symmetric. In [36, 37], the system with asymptotically linear and 3-linear nonlinearity is considered. For more results on the nonhomogeneous case, see [38, 39] and the references therein.

Throughout this paper, the letters \(C_{i}\), \(d_{i}\) will be used to denote various positive constants which may vary from line to line and are not essential to the problem. We denote by ‘⇀’ weak convergence and by ‘→’ strong convergence. Also if we take a subsequence of a sequence \(\{ u_{n}\}\), we shall denote it again \(\{u_{n}\}\). We use \(o(1)\) to denote any quantity which tends to zero when \(n\rightarrow\infty\).

Now we state our main result.

In Section 1, we know that the embedding \(E\hookrightarrow L^{s}(\mathbb {R}^{3})\) is continuous, for any \(s\in[2, 2^{\ast}]\). Furthermore, we have the following result.

We refer the reader to [3] and [11] for the details.

Now we give a proposition, which will be applied to prove Theorem 1.1. Recall that \(I\in C^{1}(E,\mathbb{R})\) is said to satisfy the \((PS)\)-condition if any sequence \(\{u_{j}\}_{j\in\mathbb{N}}\) is bounded and \(I^{\prime}(u_{j})\rightarrow0\) as \(j\rightarrow+\infty\), possesses a convergent subsequence in E.