Nonradial least energy solutions of the p-Laplace elliptic equations

Ryuji Kajikiya

doi:10.3934/dcds.2018024

Article Contents

2018, Volume 38, Issue 2: 547-561. Doi: 10.3934/dcds.2018024

This issue Previous Article Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach Next Article Energy-critical NLS with potentials of quadratic growth

Nonradial least energy solutions of the p-Laplace elliptic equations

Ryuji Kajikiya

Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan

Received: March 31, 2017

Revised: July 31, 2017

Published: November 2017

This work was supported by JSPS KAKENHI Grant Number 16K05236.

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Abstract

We study the p-Laplace elliptic equations in the unit ball under the Dirichlet boundary condition. We call u a least energy solution if it is a minimizer of the Lagrangian functional on the Nehari manifold. A least energy solution becomes a positive solution. Assume that the nonlinear term is radial and it vanishes in $|x| <a$ and it is positive in $a<|x|<1$. We prove that if a is close enough to 1, then no least energy solution is radial. Therefore there exist both a positive radial solution and a positive nonradial solution.

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Mathematics Subject Classification: Primary: 35J20, 35J25.

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