On the solutions of quasilinear elliptic equations with a polynomial-type reaction term

Abstract

We study existence and boundedness of solutions for the quasilinear elliptic equation $-\Delta_{m} u=\lambda(1+u)^p$ in a bounded domain $\Omega$ with homogeneous Dirichlet boundary conditions. The assumptions on both the parameters $\lambda$ and $p$ are fundamental. Strange critical exponents appear when boundedness of solutions is concerned. In our proofs we use techniques from calculus of variations, from critical-point theory, and from the theory of ordinary differential equations.