Multiple positive solutions for a class of $p - q$-Laplacian systems with multiple parameters and combined nonlinear effects

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 $$ \begin{cases} {-\operatorname{div} \big[ |x_N|^{-ap} | \nabla u |^{p-2} \nabla u \big] + \lambda|x_N|^{-(a+1-c)p} |u|^{p-2}u } & \\ \ \ \ = |x_N|^{-bq} |u|^{q-2} u + f & \mbox{in }\mathbb R_+^N \\ {u} = 0 &\mbox{on } \partial \mathbb R_+^N, \end{cases} $$ 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.$