## Differential and Integral Equations

### 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~$f \not\equiv 0.$

#### Article information

Source
Differential Integral Equations Volume 22, Number 7/8 (2009), 669-678.

Dates
First available in Project Euclid: 20 December 2012

Ali, Jaffar; Shivaji, R. Multiple positive solutions for a class of $p - q$-Laplacian systems with multiple parameters and combined nonlinear effects. Differential Integral Equations 22 (2009), no. 7/8, 669--678.https://projecteuclid.org/euclid.die/1356019543