Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon

Qiuping  Lu; Zhi  Ling

doi:10.3934/cpaa.2015.14.2411

Article Contents

2015, Volume 14, Issue 6: 2411-2429. Doi: 10.3934/cpaa.2015.14.2411

This issue Previous Article Symmetry of solutions to semilinear equations involving the fractional laplacian Next Article Regularity and nonexistence of solutions for a system involving the fractional Laplacian

Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon

Qiuping Lu^1, and
Zhi Ling^1,

1.
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Received: March 2015

Revised: July 2015

Published: November 2015

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Abstract

For a general elliptic problem $-\triangle{u} = g(u)$ in $R^N$ with $N \ge 3$, we show that all solutions have compact support and there exists a least energy solution, which is radially symmetric and decreases with respect to $|x| = r$. With this result we study a singularly perturbed elliptic problem $ -\epsilon^{2} \triangle{u} + |u|^{q-1}u = \lambda u + f(u)$ in a bounded domain $\Omega$ with $0 < q < 1 $, $\lambda \ge 0$ and $ u \in H^1_0(\Omega) $. For any $y \in \Omega$, we show that there exists a least energy solution $u_{\epsilon}$, which concentrates around this point $y$ as $\epsilon \to 0$. Conversely when $\epsilon $ is small, the boundary of the set $\{ x \in \Omega | u_{\epsilon}(x)>0 \}$ is a free boundary, where $u_{\epsilon}$ is any nonnegative least energy solution.

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Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 58E05.

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