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Erschienen in:
28.06.2016 | Original Research
Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in \(\mathbb {R}^{N}\)
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2017
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Abstract
In this article we study the problem where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\)
\(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.
$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$