Existence for Quasilinear Elliptic Systems with Quadratic Growth Having a Particular Structure

A. Mokrane

Abstract: In this paper, we consider the quasilinear elliptic system: $$ \left\{\eqalign{&{}-\sum_{i,j=1}^{N}\frac{\partial}{\partial x_{i}} \biggl(A_{ij}(x,u)\,\frac{\partial u^{\gamma}}{\partial x_{j}}\biggr)={} \cr &{}=G^{\gamma}(x,u,\nabla u)+F(x,u,\nabla u)\, Du^{\gamma}\hbox{in}\calc{D}'(\Omega), 1\le\gamma\le m,\cr &{}u\in(H_{0}^{1}(\Omega)\cap L^{\infty}(\Omega))^{m}.\cr}\right. $$ The right hand side of this system consists of two parts: the first one, $G^{\gamma}(x,u,\nabla u)$, can have a quadratic growth in $Du^{\delta}$ for $\delta\le\gamma$, and possibly a small quadratic growth in $Du^{\delta}$ for $\delta>\gamma$; the second part is a coupling term with the particular structure $F(x,u,\nabla u)\,Du^{\gamma}$, where the nonlinearity $F$ is the same for all the equations and can have linear growth in $\nabla u$. We approximate the problem and assume that an $L^{\infty}$-estimate on the approximated solutions is known. Without assuming any smallness on this $L^{\infty}$-estimate we then prove that the approximations converge strongly in $(H_{0}^{1}(\Omega))^{m}$ and that the system admits at least one solution.

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