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

Hirano, Norimichi; Saccon, Claudio; Shioji, Naoki

Adv. Differ. Equ. 9, No. 1-2, 197-220 (2004).

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.

MSC:

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