Existence of solution to $p-$Kirchhoff type problem in$\mathbb{R}^N$ via Nehari manifold

Caisheng Chen; Qing Yuan

doi:10.3934/cpaa.2014.13.2289

Article Contents

2014, Volume 13, Issue 6: 2289-2303. Doi: 10.3934/cpaa.2014.13.2289

This issue Previous Article Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity Next Article Mirror symmetry for a Hessian over-determined problem and its generalization

Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold

Caisheng Chen^1, and
Qing Yuan^1,

1.
College of Science, Hohai University, Nanjing, 210098

Received: August 31, 2013

Revised: December 31, 2013

Published: October 2014

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Abstract

In this paper, we study the existence of positive solution to $p-$Kirchhoff type problem \begin{eqnarray} &(a+\mu(\int_{\mathbb{R}^N}(|\nabla u|^p+V(x)|u|^p)dx)^{\tau})(-\Delta_pu+V(x)|u|^{p-2}u)=|u|^{m-2}u\\ &+\lambda |u|^{q-2}u, \; {\rm in}\; \mathbb{R}^N \\ &u(x)>0, \;\;{\rm in}\;\; \mathbb{R}^N,\;\; u\in W^{1,p}(\mathbb{R}^N), \end{eqnarray} where $a, \mu>0, \tau\ge 0, \lambda\in \mathbb{R} $ and $1 < p < N, p < q < m < p^*=\frac{pN}{N-p}$. The potential $V(x)\in C(\mathbb{R}^N)$ and $0 < \inf_{x\in\mathbb{R}^N}V(x) < \sup_{x\in\mathbb{R}^N}V(x) < \infty$. The existence of solution will be obtained by the Nehari manifold and variational method.

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Mathematics Subject Classification: 35J50, 35J75, 35J92.

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