On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities

Abstract

We deal with existence and multiplicity for the following class of nonhomogeneous Schrödinger–Poisson systems \begin{equation*} \begin{cases} -\Delta u + V(x) u + K(x) \phi(x) u = f(x, u) + g(x) \quad & \text{in } \mathbb{R}^3, \\ -\Delta \phi = K(x) u^2 \quad & \text{in } \mathbb{R}^3, \end{cases} \end{equation*} where $V, K: \mathbb{R}^3 \rightarrow \mathbb{R}^+$ are suitable potentials and $f: \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}$ satisfies sublinear growth assumptions involving a finite number of positive weights $W_i$, $i= 1,\dots,r$ with $r \geq 1$. By exploiting compact embeddings of the functional space on which we work in every weighted space $L_{W_i}^{w_i}(\mathbb{R}^3)$, $w_i \in (1, 2)$, we establish existence by means of a generalized Weierstrass theorem. Moreover, we prove multiplicity of solutions if $f$ is odd in $u$ and $g(x) \equiv 0$ thanks to a variant of the symmetric mountain pass theorem stated by R. Kajikiya for subquadratic functionals.

Keywords

• schrödinger–poisson systems
• sublinear nonlinearities
• variational methods
• compact embeddings