Multiplicity Results for Classes of Infinite Positone Problems

Abstract

We study positive solutions to the singular boundary value problem

where ${\Delta }_{p}u=$ div $(|\nabla u{|}^{p-2}\nabla u)$, $p>1,\lambda >0,\beta \in (0,1)$ and $\Omega$ is a bounded domain in ${\mathbb{R}}^N,N\ge 1.$ Here $f:[0,\infty )\to (0,\infty )$ is a continuous nondecreasing function such that ${\lim }_{u\to \infty }\frac{f(u)}{u^{\beta +p-1}}=0.$ We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha +u}}$ for $\alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-supersolutions.