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July/August 2009 Multiple positive solutions for a class of $p - q$-Laplacian systems with multiple parameters and combined nonlinear effects
Jaffar Ali, R. Shivaji
Differential Integral Equations 22(7/8): 669-678 (July/August 2009).

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.$

Citation

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Jaffar Ali. R. Shivaji. "Multiple positive solutions for a class of $p - q$-Laplacian systems with multiple parameters and combined nonlinear effects." Differential Integral Equations 22 (7/8) 669 - 678, July/August 2009.

Information

Published: July/August 2009
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35153
MathSciNet: MR2532116

Subjects:
Primary: 35J57
Secondary: 35J60, 35J62, 35J70, 47J15

Rights: Copyright © 2009 Khayyam Publishing, Inc.

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Vol.22 • No. 7/8 • July/August 2009
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