On multiple solutions generated by a parametric subcritical nonlinearity in a quasilinear elliptic Dirichlet problem

Abstract

his paper deals with the existence of weak solutions for a quasilinear Dirichlet elliptic problem with a more general subcritical reaction involving a parameter.  According to the values of the parameter, we clearly show the non-existence and multiplicity results. Specifically, we first exhibited  a critical positive value  such that  the problem  admits at least a nontrivial non-negative weak solution if and only if the parameter is greater than or equal to this critical value.  Furthermore, for the values of parameter greater than a second critical positive value, we prove the existence  of a second independent nontrivial non-negative weak solution of the problem.