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Combined effects of double nonlocal terms in the nonlinear eigenvalue problems
  • Shuai Yao,
  • Haibo Chen
Shuai Yao
Central South University School of Mathematics and Statistics

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Haibo Chen
Department of Mathematics, Central South University
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Abstract

In this paper, we study the following eigenvalue problem for Kirchhoff type equation with Hartree nonlinearity: \begin{equation} -M\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)\Delta u+\mu V(x)u=\left( I_{\alpha }\ast Q\left\vert u\right\vert^{p}\right) Q\vert u\vert ^{p-2}u+\lambda f(x)u \quad \text{in}\quad \mathbb{R}^{N}, \end{equation} where $N\geq3, a, \mu>0$ parameters, $M(t)=at+1$, $V\in C(\mathbb{R}^{N},\mathbb{R}% ^{+}) $, $I_{\alpha }$ is the Riesz potential, $Q(x)\in L^{\infty}(\mathbb{R}^{N})$ with changes sign in $\overline{\Omega}:=\left\{V(x)=0\right\}$, and $0