Positive solutions for elliptic problems with sign-changing nonlinearities
Wiktor Burakowski, Aleksandra Orpel · Discrete and Continuous Dynamical Systems - B · 2025
We discuss the class of nonlinear elliptic equations $ \Delta u(x) + f(x, u(x)) + g(x, u)x \cdot abla u(x) = 0 $ in an exterior domain, and investigate the existence of positive solutions such that $ \underset {||x||\rightarrow \infty}{\lim}u(x) = 0 $. We consider two cases: when the nonlinearity $ f $ is nonnegative, in particular $ f(x, 0)\geq0 $ ('positone problem'), as well as when $ f $ may change its sign. The latter case contains also problems with $ f $ such that $ f(x, 0)<0 $ ('semipositone problem'). Applying the subsolution and supersolution methods based on the Noussair-Swanson theorem, we show the existence of positive solutions of the main problem and discuss their properties. Our approach allows us to consider problems with $ f $ and $ g $ that are not radially symmetric.