### Simplicity of the principal eigenvalue for indefinite quasilinear problems

#### 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 $q < p$.

#### Article information

Source
Adv. Differential Equations, Volume 12, Number 4 (2007), 407-434.

Dates
First available in Project Euclid: 18 December 2012

Mathematical Reviews number (MathSciNet)
MR2305874

Zentralblatt MATH identifier
1158.35069

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 49R50

#### Citation

Kawohl, Bernd; Lucia, Marcello; Prashanth, S. Simplicity of the principal eigenvalue for indefinite quasilinear problems. Adv. Differential Equations 12 (2007), no. 4, 407--434. https://projecteuclid.org/euclid.ade/1355867457