一类带临界指标的非自治Kirchhoff型方程非平凡解的存在性

Abstract

Abstract:

We are concerned with the following non-autonomous Kirchhoff type equation in $ {{\Bbb R}} ^{3} $ $ \begin{eqnarray*} M\left(\int_{{{\Bbb R}} ^{N}}|\nabla u(x)|^{2}+\int_{{{\Bbb R}} ^{N}}V(x)|u(x)|^{2}\right) \left(-\Delta u+V(x)u \right)=\lambda K(x)f(u)+u^{5}, \end{eqnarray*} $ with vanishing potentials $ V(x), K(x) $ at infinity and a Sobolev critical term $ u^{5} $, where $ \lambda\geq 0 $ is a parameter. We prove that there exists $ \lambda^*>0 $ such that the equation has a nontrivial solution $ u_{\lambda} $ for all $ \lambda\geq\lambda^* $.

Key words: Kirchhoff type equation, Critical growth, Variational methods

CLC Number:

O175.2

Penghui Zhang,Zhiqing Han. Existence of Nontrivial Solutions for Non-autonomous Kirchhoff-type Equations with Critical Growth in $ {{\Bbb R}} ^{3} $[J].Acta mathematica scientia,Series A, 2022, 42(5): 1424-1432.

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References 20

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Existence of Nontrivial Solutions for Non-autonomous Kirchhoff-type Equations with Critical Growth in $ {{\Bbb R}} ^{3} $

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