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Keywords:
$p$-Laplacian; positive solutions; sub- and supersolutions; local minimizers; Palais-Smale condition
$p$-Laplacian; positive solutions; sub- and supersolutions; local minimizers; Palais-Smale condition
Summary:
We show that, under appropriate structure conditions, the quasilinear Dirichlet problem $$ \cases -\operatorname{div}(|\nabla u|^{p-2}\nabla u) =f(x,u), \quad & x\in\Omega, \ u=0, & x\in\partial\Omega, \endcases $$ where $\Omega $is a bounded domain in $\Bbb R^n$, $1<p<+\infty $, admits two positive solutions $u_{0}$, $u_{1}$ in $W_{0}^{1,p}(\Omega)$ such that $0<u_{0}\leq u_{1}$ in $\Omega $, while $u_{0}$ is a local minimizer of the associated Euler-Lagrange functional.
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