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Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree... This paper is concerned with the following Kirchhoff-type equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$ - ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + μ ϕ | u | p - 2 u = f ( x , u ) + g ( x , u ) , in R 3 , ( - Δ ) α 2 ϕ = μ | u | p , in R 3 , where $$a>0,~b,~\mu \ge 0$$ a > 0 , b , μ ≥ 0 are constants, $$\alpha \in (0,3)$$ α ∈ ( 0 , 3 ) , $$p\in [2,3+2\alpha )$$ p ∈ [ 2 , 3 + 2 α ) , the potential V(x) may be unbounded from below and $$\phi |u|^{p-2}u$$ ϕ | u | p - 2 u is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
DOI
10.1007/s00009-018-1170-4
Publisher site
See Article on Publisher Site

Abstract

This paper is concerned with the following Kirchhoff-type equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$ - ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + μ ϕ | u | p - 2 u = f ( x , u ) + g ( x , u ) , in R 3 , ( - Δ ) α 2 ϕ = μ | u | p , in R 3 , where $$a>0,~b,~\mu \ge 0$$ a > 0 , b , μ ≥ 0 are constants, $$\alpha \in (0,3)$$ α ∈ ( 0 , 3 ) , $$p\in [2,3+2\alpha )$$ p ∈ [ 2 , 3 + 2 α ) , the potential V(x) may be unbounded from below and $$\phi |u|^{p-2}u$$ ϕ | u | p - 2 u is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.

Journal

Mediterranean Journal of Mathematics – Springer Journals

Published: May 28, 2018

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