Generalized Quasilinear Elliptic Equations in {mathbb {R}}^N

Abstract

In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation -div(a(x,u,∇u))+At(x,u,∇u)+V(x)|u|p-2u=g(x,u)inRN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} - \ extrm{div} (a(x,u,\ abla u)) + A_t(x,u,\ abla u) + V(x) {\\vert u \\vert }^{p-2} u= g(x,u) \\quad \\quad \\hbox { in }{{\\mathbb {R}}}^{N} \\end{aligned}$$\\end{document}with p>1, Nge 2 and V:{mathbb {R}}^Nrightarrow {mathbb {R}} suitable measurable positive function. Here, we suppose A: {mathbb {R}}^N times {mathbb {R}}times {mathbb {R}}^N rightarrow {mathbb {R}} is a given {C}^{1}-Carathéodory function which grows as |xi |^p, with A_t(x,t,xi ) = frac{partial A}{partial t}(x,t,xi ), a(x,t,xi ) = nabla _xi A(x,t,xi ), V:{mathbb {R}}^Nrightarrow {mathbb {R}} is a suitable measurable function and g:{mathbb {R}}^N times {mathbb {R}}rightarrow {mathbb {R}} is a given Carathéodory function which grows as |xi |^q with 1<q<p. Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on A(x,t,xi ), V(x) and g(x, t), we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach. Thus, a minimization argument on bounded sets can be used to state the existence of a nontrivial weak bounded solution on an arbitrary bounded domain. Then, one nontrivial bounded solution of the given equation can be found by passing to the limit on a sequence of solutions on bounded domains. Finally, under slightly stronger hypotheses, we can able to find a positive solution of the problem.