A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential

Shun Kodama

doi:10.3934/cpaa.2017033

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2017, Volume 16, Issue 2: 671-698. Doi: 10.3934/cpaa.2017033

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A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential

Shun Kodama

2-32-2, Kami-takada, Nakano, Tokyo 164-0002, Japan

Author Bio: E-mail address, kodamashun0119@gmail.com

Received: March 31, 2016

Revised: July 31, 2016

Published: August 2017

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Abstract

In this paper we study the following non-autonomous singularly perturbed Dirichlet problem:

${\varepsilon ^2}\Delta u -u + K(x)f(u) = 0, \; u > 0\quad {\rm{in}}\; \Omega, \quad u = 0\quad {\rm{on}}\; \partial \Omega, $

for a totally degenerate potential K. Here ε > 0 is a small parameter, $\Omega \subset \mathbb{R}^N$ is a bounded domain with a smooth boundary, and f is an appropriate superlinear subcritical function. In particular, f satisfies $0 < \liminf_{ t \to 0+} f(t)/t^q \leq \limsup_{ t \to 0+} f(t)/t^q < + \infty$ for some $1 < q < + \infty$. We show that the least energy solutions concentrate at the maximal point of the modified distance function $D(x) = \min \{ (q+1) d(x, \partial A), 2 d(x, \partial \Omega) \}$, where $A = \{ x \in \bar{ \Omega } \mid K(x) = \max_{ y \in \bar{ \Omega } } K(y) \}$ is assumed to be a totally degenerate set satisfying ${{A}^{{}^\circ }}\ne \emptyset $.

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Mathematics Subject Classification: Primary: 35B25, 35B40; Secondary: 35J20.

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