MULTIPLICITY OF POSITIVE SOLUTIONS TO SCHRO\"DINGER-TYPE p-LAPLACIAN PROBLEMS
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초록
We establish a multiplicity result for positive solutions to a Schr\"odinger-type $p$-Laplacian problem:$-\Delta_p u+V(x)|u|^{p-2}u=\lambda f(u)$ in $ \Omega, ~u=0 $ on $\partial \Omega,$where $\Delta_p u= div (|\nabla u |^{p-2} | \nabla u)$ is the $p$-Laplacian operator, $p\geq 1, \Omega$ is a bounded domain in $ \mathbb{R}^N, N \geq 2, \lambda$ is a positive parameter,$V \in L^\infty(\Omega)$ and $f:[0, \infty) \rightarrow (0, \infty)$ is a continuous function. In particular, when $f$ is $p$- sublinear at infinity,we discuss the existence of at least three positive solutions for a certain range of $\lambda$. The proofs are mainly based on the sub and supersolution method.
Abstract
We establish a multiplicity result for positive solutions to a Schr\"odinger-type $p$-Laplacian problem:$-\Delta_p u+V(x)|u|^{p-2}u=\lambda f(u)$ in $ \Omega, ~u=0 $ on $\partial \Omega,$where $\Delta_p u= div (|\nabla u |^{p-2} | \nabla u)$ is the $p$-Laplacian operator, $p\geq 1, \Omega$ is a bounded domain in $ \mathbb{R}^N, N \geq 2, \lambda$ is a positive parameter,$V \in L^\infty(\Omega)$ and $f:[0, \infty) \rightarrow (0, \infty)$ is a continuous function. In particular, when $f$ is $p$- sublinear at infinity,we discuss the existence of at least three positive solutions for a certain range of $\lambda$. The proofs are mainly based on the sub and supersolution method.