Abstract
We study the existence of multiple positive solutions of $ -\Delta u = \lambda u^{-q} +u^p $ in $\Omega$ with homogeneous Dirichlet boundary condition, where $\Omega$ is a bounded domain in $\mathbb R^N$, $\lambda >0$, and $0 < q \leq 1 < p \leq (N+2)/(N-2)$. We show by a variational method that if $\lambda$ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of $q=1$ or $p=(N+2)/(N-2)$. We also study the regularity of positive weak solutions.
Citation
Norimichi Hirano. Claudio Saccon. Naoki Shioji. "Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities." Adv. Differential Equations 9 (1-2) 197 - 220, 2004. https://doi.org/10.57262/ade/1355867973