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2007 Simplicity of the principal eigenvalue for indefinite quasilinear problems
Bernd Kawohl, Marcello Lucia, S. Prashanth
Adv. Differential Equations 12(4): 407-434 (2007).

Abstract

Given any domain $\Omega \subseteq \mathbb R^N$, $w \in L^1_{loc} (\Omega)$ and a differentiable function $A: \mathbb R^N \to \mathbb [0,\infty)$ which is $p$-homogeneous and strictly convex, we consider the minimization problem $$ \inf \Big \{ \frac{\int_{\Omega} A( \nabla u ) } { \left( \int_{\Omega} w (x) |u|^q \right)^{\frac{p}{q} }} \, \colon \, u \in \mathcal{D}^{1,\mathcal{p}}_0 (\Omega) \quad 0< \int_{\Omega} w (x) |u|^q < \infty \Big \}. $$ If the infimum is achieved and $q =p >1$, without additional regularity assumptions on $\Omega$ or the weight function $w$, we show that the minimizer is unique up to a constant factor. The same conclusion holds when $A$ is allowed to depend on $x \in \Omega$ and satisfies natural growth assumptions. Some of our results also hold when $q < p$.

Citation

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Bernd Kawohl. Marcello Lucia. S. Prashanth. "Simplicity of the principal eigenvalue for indefinite quasilinear problems." Adv. Differential Equations 12 (4) 407 - 434, 2007.

Information

Published: 2007
First available in Project Euclid: 18 December 2012

zbMATH: 1158.35069
MathSciNet: MR2305874

Subjects:
Primary: 35J20
Secondary: 35J60, 49R50

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.12 • No. 4 • 2007
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