Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips

Alberto Farina; Berardino Sciunzi

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arXiv · 1701.04720

Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips

Alberto Farina·Berardino Sciunzi

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Abstract

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 $f(0)<0$, $f(0)= 0$ or $f(0)> 0$. 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 $f$.

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Alberto Farina, Berardino Sciunzi. 2017-01-17. Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips. https://arxiv.org/abs/1701.04720

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