Monotonicity and Symmetry of Nonnegative Solutions to -Δ u=f(u) in Half-Planes and Strips

Abstract

We consider nonnegative solutions to -Δ⁢u=f⁢(u)${-\Delta u=f(u)}$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating and 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 cases: f⁢(0)0${f(0)>0}$.Furthermore, we make the effort to deal with nonlinearities f that may be not locally-Lipschitz continuous.We also provide explicit examples showing the sharpness of our assumptions on the nonlinear function f.

Keywords

• semilinear elliptic equations
• qualitative properties of the solutions
• moving plane method
• 35j61
• 35b51
• 35b06