Positive solutions for parametric Dirichlet problems with indefinite potential and superdiffusive reaction

Abstract

We consider a parametric semilinear Dirichlet problem driven by the Laplacian plus an indefinite unbounded potential and with a reaction of superdifissive type. Using variational and truncation techniques, we show that there exists a~critical parameter value $\lambda_{\ast} \gt 0$ such that for all $\lambda \gt \lambda_{\ast}$ the problem has at least two positive solutions, for $\lambda= \lambda_{\ast}$ the problem has at least one positive solution, and no positive solutions exist when $\lambda\in( 0,\lambda_{\ast})$. Also, we show that for $\lambda\geq$ $\lambda_{\ast}$ the problem has a~smallest positive solution.