Existence and multiplicity of systems of Kirchhoff‐type equations with general potentials

Abstract

This paper is concerned with the following systems of Kirchhoff‐type equations: −()a+b∫double-struckRN|∇u|2dxnormalΔu+V(x)u=Fu(x,u,v),x∈double-struckRN,−()c+d∫double-struckRN|∇v|2dxnormalΔv+V(x)v=Fv(x,u,v),x∈double-struckRN,u(x)→0,1em1em1em1emv(x)→01em1em1em1em1em1em1em1em1em1em1em1em1em1em1emas1em|x|→∞. Under more relaxed assumptions on V(x) and F(x,u,v), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd.

Ground state solutions for the nonlinear Schrödinger‐Poisson systems with sum of periodic and vanishing potentials

“…See, for example, the literature. () Let us briefly recall some known results on system . In previous study, D'Aprile and Mugnai considered for the case of f ( x , u )=| u | p −2 u and V ( x )=1, i.e., …”

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“…In [27,30], Z. Zhang et al studied the existence of nontrivial solutions and sign-changing solutions for (1). The study of systems of Kirchhoff type equations can be found in [6,8,13,14,29,32]. Critical problems involving nonlocal operators creat many difficulties in applying variational methods.…”