Multiple solutions for elliptic systems via trapping regions and related nonsmooth potentials

Abstract

This paper deals with the Dirichlet boundary value problem for quasilinear elliptic systems in a bounded domain with a diagonal -Laplacian as leading differential operator and a Carathéodory right-hand side vector field . Only by imposing certain growth conditions on , , near zero we are able to prove the existence of multiple, nontrivial solutions, and provide sign information for them. More precisely, first we show the existence of a positive minimal and a negative maximal solution, where the notion maximal and minimal refer to the partial ordering of vector-valued functions introduced by the order cone . Second, in case the considered system is, in addition, of variational structure, i.e. with sufficiently smooth, further nontrivial solutions can be proved to exist, in particular, solutions that change sign. Our approach is based, on the one hand, on a sequences of expanding trapping regions to get extremal constant sign solutions. On the other hand, introducing certain truncation operators we construct a (nonsmooth) functional whose critical points turn out to be solutions of the given system within the trapping region formed by the extremal solutions. A characteristic feature of this approach is that one cannot avoid to deal with nonsmooth potentials no matter how smooth might be. Further tools used to achieve our goals are comparison results for differential inequalities, and variational and topological tools for nonsmooth functionals such as, e.g. nonsmooth critical point theory and second deformation lemma for nonsmooth, locally Lipschitz functionals. It is also worth pointing out that in the study of system we essentially use knowledge on associated elliptic equations.