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March  2017, 16(2): 671-698. doi: 10.3934/cpaa.2017033

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

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

Received  April 2016 Revised  August 2016 Published  January 2017

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 ${{A}^{{}^\circ }}\ne \emptyset$.
Citation: Shun Kodama. A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential. Communications on Pure & Applied Analysis, 2017, 16 (2) : 671-698. doi: 10.3934/cpaa.2017033
References:

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References:
Example 2 (ⅰ)
Example 2 (ⅱ)
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