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.$