2006 International Conference in honor of Jacqueline Fleckinger. Electron. J. Diff. Eqns., Conference 16 (2007), pp. 29-34.

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

Jaffar Ali, R. Shivaji

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

Published May 15, 2007.
Math Subject Classifications: 35J55, 35J70.
Key Words: Positive solutions; p-Laplacian systems; semipositone problems.

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Jaffar Ali
Department of Mathematics
Mississippi State University
Mississippi State, MS 39759, USA
email: js415@ra.msstate.edu
Ratnasingham Shivaji
Department of Mathematics
Mississippi State University
Mississippi State, MS 39759, USA
email: shivaji@ra.msstate.edu

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