## Differential and Integral Equations

### Multiplicity results for a degenerate quasilinear elliptic equation in half-space

#### Abstract

In this work, we prove a multiplicity result for a class of quasilinear elliptic equation involving the subcritical Hardy-Sobolev exponent, and singularities both in the operator and in the non-linearity. Precisely, we study the problem $\left\{\begin{array}{rcll} -\mbox{div} & = & [|x_N|^{-ap}|\nabla_u|^{p-2}\nabla u]+\lambda|x_N|^{-(a+1-c)p}|u|^{p-2}u \\ & = & |x_N|^{-ap}|u|^{q-2}u+f & \mbox{in }\mathbb{R}^N_+ \\ u & = & 0 & \mbox{on } \partial\mathbb{R}^N_+ \end{array}\right.$where we denote $x=(x_1,x_2,\dots,x_N)=(x',x_N) \in \mathbb R^{N-1}\times \mathbb R$, $\mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N > 0 \right\}$, $\partial \mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N = 0 \right\}$, and we consider $1 < p < N$, $0 \leqslant a < (N-p)/p$, $a < b < a+1$, $c=0$, $d \equiv a+1-b$, $q = q(a,b) \equiv Np/(N - pd)$ (the Hardy-Sobolev critical exponent), $\lambda \in \mathbb R$ is a parameter, and $f \in \big( L_b^q(\mathbb R_+^N) \big)^{*}$, the dual space of the weighted Lebesgue space. We prove an existence result for the case $f \equiv 0$ and a multiplicity result in the case $\lambda = 0$ for non-autonomous perturbations~$f \not\equiv 0.$

#### Article information

Source
Differential Integral Equations, Volume 22, Number 7/8 (2009), 753-770.

Dates
First available in Project Euclid: 20 December 2012