A quasilinear elliptic system with natural growth terms
The interesting paper under review deals with elliptic systems with quadratic growth gradient terms of the type \[ \begin{cases} -\text{div\,}\big(a(x,z)\nabla u)\big)=f(x) & \text{in}\;\Omega,\\ -\text{div\,}\big(b(x)\nabla z)\big)+h(x,z)|\nabla u|^2=g(x) & \text{in}\;\Omega,\\ u=z=0 & \text{on}\;\partial\Omega, \end{cases} \] on a bounded domain \(\Omega\subset \mathbb{R}^N,\) \(N>2,\) where \(a(x,s),\;h(x,s): \Omega\times\mathbb{R}\to \mathbb{R}\) are Carathéodory functions, \(b:\Omega\to\mathbb{R}\) is measurable, \[ \alpha\leq a(x,s),\quad \alpha\leq b(x)\leq \beta,\quad h(x,s)\cdot s\geq0 \] with \(0<\alpha\leq \beta,\) while \(f\) and \(g\) belong to suitable Lebesgue spaces. First of all, the authors prove existence of solutions to the above problem under the assumption \(a\in L^\infty\) which allows them to deal with \(f,g\in L^1(\Omega).\) After that, the case of unbounded \(a\) is studied when the lower-order term \(h(x,s)\) grows suitably with respect to \(a.\) It is shown how the presence of a quadratic gradient term in the second equation ``breaks'' the maximum principle: if \(f\in L^1(\Omega)\) then the solution \(z\) of the second equation has to be zero on the set of ``explosion'' of \(f\) even if \(g\) is strictly positive in \(\Omega.\) The case of measure data, yielding nonexistence of solutions is studied as well.
quasilinear quadratic elliptic equations
scientific article; zbMATH DE number 6511871
scientific article