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 $\mathcal{C}^{1}$-Carath\'eodory functions $A, B:\Omega\times\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}$ 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 $\mathcal{C}^{1}$-Carath\'eodory 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.