## 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.

### 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
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