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Ground state solutions and least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in \(\mathbb R^N\). (English) Zbl 1357.35115

Summary: In this article we study the problem \[ \Delta^2u-\left(1+\lambda\int_{\mathbb R^N}|\nabla u|^2dx\right)\Delta u+V(x)u=|u|^{p-2}u\text{ in }\mathbb R^N, \] where \(\Delta^2:=\Delta(\Delta)\) is the biharmonic operator, \(\lambda>0\) is a parameter, \(p\in(2, 2_\ast)\), and \(V(x)\in C(\mathbb R^N,\mathbb R)\). Under appropriate assumptions on \(V(x)\), the existence of ground state solutions and a least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

MSC:

35J35 Variational methods for higher-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
35J60 Nonlinear elliptic equations
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