Positive solutions for a Schrödinger–Poisson system involving concave–convex nonlinearities

On a Schrödinger–Poisson system with singularity and critical nonlinearities

“…System (1.2) has been extensively studied, focusing on the existence of positive solutions, multiplicity of solutions, ground state solutions, sign-changing solutions, radial solutions, by using the variational methods and critical point theory under various assumptions of potential V and nonlocal term f , see for example [4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein. In addition, existence and multiplicity of the Schrödinger-Poisson problem in a bounded domain has been paid attention to by many authors, we can see [18][19][20][21][22][23][24]. More precisely, Fan [21] considered the following system:…”

“…For η = 1 and λ > 0, the author obtained the existence and uniqueness of positive solution of system (1.3) by using variational method; for η = -1 and λ > 0 small enough, the author also considered the existence and multiplicity of positive solutions via Nehari manifold. For the case that replaced with concave-convex nonlinearities and critical growth terms of system (1.3), the authors in [23] got two positive solutions by using the variational method and the concentration-compactness principle when λ is small enough. Recently, Zheng [24] studied the following Schrödinger-Poisson system:…”

Existence of positive ground state solutions of Schrödinger–Poisson system involving negative nonlocal term and critical exponent on bounded domain

“…This implies that J is empty, which means that Ω |u n | 6 dx → Ω |u| 6 dx. We can also get u n → u in X (see Lemma 2.2 in [8]). So Lemma 3.2 holds.…”

“…For simplicity, in many cases, we just say that u ∈ X, instead of (u, φ u ) ∈ X × X, is a weak solution of system (1.1). It is easy to see that I ∈ C 1 (X, R) (see [8,9]) and…”

“…The Schrödinger-Poisson system on whole space R N has attracted a lot of attention. Few works concern the existence of solutions for the Schrödinger-Poisson system on a bounded domain, particularly, critical nonlinearity except [2,7,8]. Up to now, Schrödinger-Poisson system (1.1) has never been studied by variational methods.…”