Abstract
In this paper, we prove the existence of positive ground state solutions of the Schrödinger–Poisson system involving a negative nonlocal term and critical exponent on a bounded domain. The main tools are the mountain pass theorem and the concentration compactness principle.
Highlights
In this paper, we consider the following Schrödinger–Poisson system:⎧ ⎪⎪⎨– u – φu = λuq–1 + u5 in Ω, ⎪⎪⎩–u φ =φ= u2 =0 in Ω, on ∂Ω, (1.1)where λ > 0 is a parameter, 2 < q < 6, and Ω is a smooth bounded domain in R3
It is standard to see that system (1.1) is variational and its solutions are the critical points of the functional defined in X by
Is an extremal function for the minimum problem (2.1), that is, it is a positive solution of the equation
Summary
Few works concern the existence of solutions for the Schrödinger–Poisson system on a bounded domain, critical nonlinearity except [2, 7, 8]. Zhang [19] considered the negative nonlocal Schrödinger–Poisson system on a bounded domain and obtained thtat there are at least two solutions involving a singularity term by using the Nehari method. There exists λ∗ > 0 such that system (1.1) has at least one positive ground state solution for all λ > λ∗. The Lax–Milgram theorem implies that for all u ∈ X, there exists a unique φu ∈ X such that
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