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On positive solutions for a class of strongly coupled p-Laplacian systems

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Author(s): Jaffar Ali | R. Shivaji

Journal: Electronic Journal of Differential Equations
ISSN 1072-6691

Volume: Conference;
Issue: 16;
Start page: 29;
Date: 2007;
Original page

Keywords: Positive solutions | p-Laplacian systems | semipositone problems.

ABSTRACT
Consider the system $$displaylines{ -Delta_pu =lambda f(u,v)quadhbox{in }Omegacr -Delta_qv =lambda g(u,v)quadhbox{in }Omegacr u=0=v quad hbox{on }partialOmega }$$ where $Delta_sz=hbox{m div}(| abla z|^{s-2} abla z)$, $s>1$, $lambda$ is a non-negative parameter, and $Omega$ is a bounded domain in $mathbb{R}$ with smooth boundary $partialOmega$. We discuss the existence of a large positive solution for $lambda$ large when $$ lim_{xoinfty}frac{f(x,M[g(x,x)]^{1/q-1})}{x^{p-1}}=0 $$ for every $M>0$, and $lim_{xoinfty} g(x,x)/x^{q-1}=0$. In particular, we do not assume any sign conditions on $f(0,0)$ or $g(0,0)$. We also discuss a multiplicity results when $f(0,0)=0=g(0,0)$.

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