Electron. J. Differential Equations, Vol. 2019 (2019), No. 13, pp. 1-16.

Existence of infinitely many small solutions for sublinear fractional Kirchhoff-Schrodinger-Poisson systems

Jose Carlos de Albuquerque, Rodrigo Clemente, Diego Ferraz

We study the Kirchhoff-Schrodinger-Poisson system
 m([u]_{\alpha}^2)(-\Delta)^\alpha u+V(x)u+k(x)\phi u = f(x,u), \quad
 (-\Delta)^\beta \phi = k(x)u^2, \quad x\in\mathbb{R}^3,
where $[\cdot]_{\alpha}$ denotes the Gagliardo semi-norm, $(-\Delta)^{\alpha}$ denotes the fractional Laplacian operator with $\alpha,\beta\in (0,1]$, $4\alpha+2\beta\geq 3$ and $m:[0,+\infty)\to[0,+\infty)$ is a Kirchhoff function satisfying suitable assumptions. The functions V(x) and k(x) are nonnegative and the nonlinear term f(x,s) satisfies certain local conditions. By using a variational approach, we use a Kajikiya's version of the symmetric mountain pass lemma and Moser iteration method to prove the existence of infinitely many small solutions.

Submitted June 25, 2018. Published January 25, 2019.
Math Subject Classifications: 35A15, 35J50, 35R11, 45G05.
Key Words: Kirchhoff-Schrodinger-Poisson equation; fractional Laplacian; variational method.

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José Carlos de Albuquerque
Institute of Mathematics and Statistics
Federal University of Goiás
74001-970, Goiânia, Goiás, Brazil
email: joserre@gmail.com
Rodrigo Clemente
Department of Mathematics
Rural Federal University of Pernambuco
52171-900, Recife, Pernambuco, Brazil
email: rodrigo.clemente@ufrpe.br
Diego Ferraz
Department of Mathematics
Federal University of Rio Grande do Norte
59078-970, Natal, Rio Grande do Norte, Brazil
email: diego.ferraz.br@gmail.com

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