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Abstract

In this paper, we study the following Schrödinger–Poisson system $\begin{matrix} \{\begin{matrix} - Δ u + u + λ ϕ u = f (u) + u^{3}, & x \in R^{4}, \\ - Δ ϕ = u^{2}, & x \in R^{4}, \end{matrix} \end{matrix}$ where $λ > 0$ is a parameter, $f \in C (R)$ and the nonlinear growth of $u^{3}$ reaches the Sobolev critical exponent since $2^{*} = 4$ in dimension 4. Under some suitable assumptions on f, we establish the existence of a positive radial and non-radial ground state solutions for the above system by using variational methods. We also discuss the asymptotic behavior of solutions with respect to the parameter λ.

Keywords

Schrödinger–Poisson system variational method critical exponent ground state solution

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Positive ground state solutions for a class of Schrödinger–Poisson systems in R 4 involving critical Sobolev exponent

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