Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips
Alberto Farina, Berardino Sciunzi · arXiv (Cornell University) · 2017
We consider nonnegative solutions to $-Δu=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity $f$. In fact we provide a unified approach that works in all the cases $f(0) 0$. Furthermore we make the effort to deal with nonlinearities $f$ that may be not locally-Lipschitz continuous. We also provide explicite examples showing the sharpness of our assumptions on the nonlinear function $f$.