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Infinitely many solutions of systems of Kirchhoff-type equations with general potentials. (English) Zbl 1408.35035

Summary: This paper is concerned with the following systems of Kirchhoff-type equations: \[ \begin{cases} -\bigg(a+b\int_{\mathbb{R}^{N}}|\nabla u|^2\mathrm{d}x\bigg)\Delta u \\ +V(x)u=F_u(x,u,v) & x\in \mathbb{R}^{N},\\ -\bigg(c+d\int_{\mathbb{R}^{N}}|\nabla v|^2\mathrm{d}x\bigg)\Delta v \\ +V(x)v=F_v(x,u,v) & x\in \mathbb{R}^{N},\\ u(x)\rightarrow 0, v(x)\rightarrow 0 \,\,\text{as}\,\,|x|\rightarrow\infty. \end{cases} \] Under some more relaxed assumptions on \(V(x)\) and \(F(x,u,v)\), we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are greatly improved and extended.

MSC:

35J60 Nonlinear elliptic equations
35J50 Variational methods for elliptic systems
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References:

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