×

Existence for quasilinear elliptic systems with quadratic growth having a particular structure. (English) Zbl 0897.35031

Summary: We consider the quasilinear elliptic system: \[ \begin{split}-\sum^N_{i,j=1} {\partial \over \partial x_i} \left(A_{ij} (x,u) {\partial u^\gamma \over \partial x_j} \right)=\\ =G^\gamma (x,u, \nabla u)+ F(x,u,\nabla u) Du^\gamma \text{ in } {\mathcal D}' (\Omega),\;1\leq \gamma\leq m, \quad u\in \bigl(H^1_0 (\Omega)\cap L^\infty (\Omega)\bigr)^m. \end{split} \] 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 \(\delta\leq\gamma\), 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 \((H^1_0 (\Omega))^m\) and that the system admits at least one solution.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A35 Theoretical approximation in context of PDEs
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
PDFBibTeX XMLCite
Full Text: EuDML