Journal of Convex Analysis 32 (2025), No. 4, 1117--1134
Copyright Heldermann Verlag 2025
Schrödinger-Poisson System Involving Potential Vanishing at Infinity and Unbounded Below
Genivaldo P. Correa
Faculty of Exact and Technological Sciences, Federal University of Pará, Abaetetuba, Brazil
genivaldo@ufpa.br
Gelson C. G. dos Santos
Institute of Exact and Natural Sciences, Federal University of Pará, Belém, Brazil
gelsonsantos@ufpa.br
[Abstract-pdf]
This article concerns the following class of system \begin{equation*} \left\{ \begin{array}{lr} -\Delta u +V(x)u+\ell(x)\phi u = f(u) + \lambda|u|^{q-2}u & \mbox{in } \mathbb{R}^3,\\[2mm] -\Delta \phi = \ell(x)u^{2} & \mbox{in } \mathbb{R}^3,\\[2mm] u,\phi\in D^{1,2}(\mathbb{R}^3), \ u,\phi\geq0 & \mbox{in } \mathbb{R}^3, \end{array} \right. \end{equation*} where $\lambda\geq0$ and $q\geq2^*=6$ is the critical Sobolev exponent in dimension 3, the nonlinearity $f:\mathbb{R}\rightarrow \mathbb{R}$ is superlinear and has subcritical growth, $V,\ell: \mathbb{R}^3\rightarrow \mathbb{R}$ are measurable functions with $\ell\in L^2(\mathbb{R}^3)$, the potential $V$ can change sign in $\mathbb{R}^3$ and vanish at infinity, that is, $V (x) \rightarrow 0 $ as $|x|\rightarrow\infty$. Our approach is based on variational method combined with Benci-Fortunato's reduction argument [\,Topol.\ Methods Nonlinear Anal.\ 11 (1998) 283--293], Del Pino-Felmer's penalization technique [\,Calc. Var. Partial Diff. Equations 4 (1996) 121--137] and $L^\infty$-estimate.
Keywords: Schroedinger-Poisson system, variational methods, Mountain Pass Theorem, nontrivial solution, supercritical exponents, vanishing potential, sign-changing potential.
MSC: 35A15, 35Q61, 35B38, 35B09.
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