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 \begin{equation*} \left\{ \begin{array}{ll} - {\rm div} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &\hbox{ in $\Omega$,}\\[10pt] - {\rm div} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left(x, u, v\right) &\hbox{ in $\Omega$,}\\[10pt] u = v = 0 &\hbox{ on $\partial \Omega$,} \end{array} \right. \end{equation*} where $\Omega$ is an open bounded domain in ${\mathbb{R}}^N$, $N\ge 2$. We suppose that some ${\mathcal{C}}^1$-Carathéodory functions $A,B:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ 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 $G:\Omega \times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$. Roughly speaking, we assume that $A(x,t,\xi )$ grows at least as $(1+|t{|}^{s_1{p}_1})|\xi {|}^{p_1}$, $p_1>1$, $s_1\ge 0$, while $B(x,t,\xi )$ grows as $(1+|t{|}^{s_2{p}_2})|\xi {|}^{p_2}$, $p_2>1$, $s_2\ge 0$, 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.

References in corpus (3)

• Bounded solutions for quasilinear modified Schrödinger equations
• Existence of radial bounded solutions for some quasilinear elliptic equations in R^N
• Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type