Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth

Abstract

In this paper, we consider the following coupled gradient-type quasilinear elliptic system -div(a(x,u,∇u))+At(x,u,∇u)=Gu(x,u,v)inΩ,-div(b(x,v,∇v))+Bt(x,v,∇v)=Gvx,u,vinΩ,u=v=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{*{20}l} - {\ ext{div}} ( a(x, u, \ abla u) ) + A_t (x, u, \ abla u) = G_u(x, u, v) &{}{\\hbox { in }}\\Omega ,\\\\ - {\ ext{div}} ( b(x, v, \ abla v) ) + B_t(x, v, \ abla v) = G_v\\left( x, u, v\\right) &{}{\\hbox { in }}\\Omega ,\\\\ u = v = 0 &{}{\\hbox { on }}\\partial \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where Omega is an open bounded domain in {mathbb {R}}^N, Nge 2. We suppose that some mathcal {C}^{1}–Carathéodory functions A, B:Omega times {mathbb {R}}times {mathbb {R}}^Nrightarrow {mathbb {R}} exist such that a(x,t,xi ) = nabla _{xi } A(x,t,xi ), A_t(x,t,xi ) = frac{partial A}{partial t} (x,t,xi ), b(x,t,xi ) = nabla _{xi } B(x,t,xi ), B_t(x,t,xi ) =frac{partial B}{partial t}(x,t,xi ), and that G_u(x, u, v), G_v(x, u, v) are the partial derivatives of a mathcal {C}^{1}–Carathéodory nonlinearity G:Omega times {mathbb {R}}times {mathbb {R}}rightarrow {mathbb {R}}. Roughly speaking, we assume that A(x,t,xi ) grows at least as (1+|t|^{s_1p_1})|xi |^{p_1}, p_1 > 1, s_1 ge 0, while B(x,t,xi ) grows as (1+|t|^{s_2p_2})|xi |^{p_2}, p_2 > 1, s_2 ge 0, and that G(x, u, v) can also have a supercritical growth related to s_1 and s_2. Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.