Existence of positive solutions for a Schrödinger-Poisson system with bounded potential and weighted functions in \(\mathbb{R}^{3}\)

Such a system, also known as the Schrödinger-Maxwell system, arises in many fields of physics. For example, the Schrödinger-Poisson system can describe the interaction of a charged particle with its own electrostatic field in quantum mechanics. The unknowns u and ϕ represent the wave functions associated with the particle and electric potential, and the functions V and K are, respectively, an external potential and nonnegative density charge. We refer to Benci and Fortunato [2] for more details on the physical aspects. This model can also appear in semiconductor theory to describe solitary waves [3].

Liu et al. [14] also considered the existence of a solution for problem (1.2) with the potential \(V(x)\in C(\mathbb{R}^{3})\) satisfying \(\inf_{x\in\mathbb{R}^{3}}V(x)>-\infty\) and for every \(M>0\), \(\operatorname{meas}(\{x\in\mathbb{R}^{3} |V(x)\le M\})<\infty\). It is well known that this assumption guarantees that the embedding \(W^{1,2}(\mathbb{R}^{3})\hookrightarrow L^{p}(\mathbb{R}^{3})\) is compact for each \(2\le p<6\). For problem (1.2), the function \(f(x,u)\) verifies \(uf(x,u)\ge4F(x,u)\) with \(F(x,u)=\int_{0}^{u}f(x,t)dt\). Similar assumptions can be found in [15–19].

Clearly, \(V(x)\) is not necessarily radial and coercive. For these assumptions, the embedding \(W^{1,2}(\mathbb{R}^{3})\hookrightarrow L^{p}(\mathbb{R}^{3})\) is not compact. Furthermore, for problem (1.1), the function \(f(x,u)\) fails to satisfy the assumption \(uf(x,u)\ge4F(x,u)\). So the variational technique for problem (1.1) becomes more delicate. Arguing as in [20, 21], to preserve this compactness in some extent for our problem, we split a minimizing sequence \(\{u_{n}\}\) into two parts: \(u_{n}=u'_{n}+u''_{n}\) (\(n\in \mathbb{N}\)) such that \(u'_{n} \to u, u''_{n}\to0\) in \(L^{m}(\mathbb{R}^{3},a)\cap L^{q}(\mathbb{R}^{3},b)\). We will obtain a positive solution by using the Nehari manifold method.

In general, let \(\|u\|_{p,\rho}=(\int_{\mathbb {R}^{3}}\rho|u|^{p}\,dx)^{1/p}\) with \(p\ge1\) and \(\rho=\rho(x)\ge0, \neq0\) a.e. in \(\mathbb {R}^{3}\). In particular, denote \(\|u\|_{p}=(\int_{\mathbb {R}^{3}}|u|^{p}\,dx)^{1/p}\) or \(\|u\|_{L^{p}(\Omega)}=(\int_{\Omega}|u|^{p}\,dx)^{1/p}\) with the domain \(\Omega\subset \mathbb {R}^{3}\).

Hence, if \(u\in E\) is a critical point of J, that is, \(J'(u)v=0\) for \(\forall v\in E\), then the pair \((u,\phi_{u})\) is a solution of problem (1.1)(see [4, 23]). For the sake of simplicity in many cases, we just say that \(u\in E\), instead of \((u,\phi_{u})\in E\times\mathcal{D}^{1,2}(\mathbb {R}^{3})\), is a weak solution of problem (1.1).

Our main result in this paper is as follows.

This paper is organized as follows. In Section 2, we set up the variational framework and establish some lemmas, which will be used in the proof of Theorem 1.1. In Section 3, we prove Theorem 1.1.

In this section, we are going to establish a series of lemmas to prove Theorem 1.1, in which we tacitly assume that the conditions in Theorem 1.1 are satisfied. We first set up the variational framework for problem (1.1).

Let \(J(u):E\rightarrow\mathbb{R} \) be the energy functional associated with problem (1.1) defined by (1.16), and its Gateaux derivative is given by (1.18).

Since u satisfies (3.1) and (3.2), it follows that \(u\in\mathcal{N}\) and so that \(J(u)=d\) and u is a minimum point for J on \(\mathcal{N}\).

To finish the proof of Theorem 1.1, it is sufficient to prove that u is a critical point for the functional \(J(u)\) in E, that is, \(J'(u)v=0\) for all \(v\in E\), and thus \(J'(u)=0\) in \(E^{*}\).