Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. (English) Zbl 1387.35287

Summary: 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 \(0<q\leq1<p\leq(N+2)/(N-2)\). 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.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
47J30 Variational methods involving nonlinear operators
Full Text: Euclid