Infinite multiplicity for an inhomogeneous supercritical problem inentire space

Liping Wang; Juncheng Wei

doi:10.3934/cpaa.2013.12.1243

Article Contents

2013, Volume 12, Issue 3: 1243-1257. Doi: 10.3934/cpaa.2013.12.1243

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Infinite multiplicity for an inhomogeneous supercritical problem in entire space

Liping Wang^1, and
Juncheng Wei^2,

1.
Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
2.
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received: December 2011

Revised: July 2012

Published: May 2013

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Abstract

Let $K(x)$ be a positive function in $R^N, N \geq 3$ and satisfy $\lim\limits_{|x|\rightarrow \infty} K(x) = K_\infty$ where $K_\infty$ is a positive constant. When $p > \frac{N + 1}{N - 3}, N \geq 4$, we prove the existence of infinitely many positive solutions to the following supercritical problem: \begin{eqnarray*} \Delta u(x) + K(x)u^p = 0, u>0 \quad in \quad R^N, \lim_{|x|\rightarrow \infty} u(x) = 0. \end{eqnarray*} If in addition we have, for instance, $\lim\limits_{|x| \rightarrow \infty}|x|^\mu (K(x) - K_\infty ) = C_0 \neq 0, 0 < \mu \leq N - \frac{2p+2}{p-1}$, then this result still holds provided that $p > \frac{N + 2}{N - 2}$.

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Mathematics Subject Classification: Primary 35B35, 92C15; Secondary 35B40, 92D25.

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