Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials

Norihisa Ikoma

doi:10.3934/dcds.2015.35.943

Article Contents

2015, Volume 35, Issue 3: 943-966. Doi: 10.3934/dcds.2015.35.943

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Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials

Norihisa Ikoma^1,

1.
Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai-Shi, Miyagi 980-8578

Received: January 31, 2014

Revised: April 30, 2014

Published: February 2015

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Abstract

In this paper, we study the existence of ground state solutions to the nonlinear Kirchhoff type equations \[ - m \left( \| \nabla u \|_{L^2(\mathbf{R}^N)}^{2} \right) \Delta u + V(x) u = |u|^{p-1} u \quad {\rm in}\ \mathbf{R}^N, \ u \in H^1(\mathbf{R}^N), \ N \geq 1 \] where $ 1 < p < \infty$ when $N=1,2$, $1 < p < (N+2)/(N-2)$ when $N \geq 3$, $m: [0,\infty) \to (0,\infty)$ is a continuous function and $V:\mathbf{R}^N \to \mathbf{R}$ a smooth function. Under suitable conditions on $m(s)$ and $V$, it is shown that a ground state solution to the above equation exists.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35B99.

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