Positive solutions for elliptic problems with sign-changing nonlinearities

Wiktor Burakowski; Aleksandra Orpel

doi:10.3934/dcdsb.2025050

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2025, Volume 30, Issue 11: 4276-4286. Doi: 10.3934/dcdsb.2025050

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Positive solutions for elliptic problems with sign-changing nonlinearities

Wiktor Burakowski^1, , and
Aleksandra Orpel^2,

1.
Doctoral School of Exact and Natural Science, University of Lodz, Poland
2.
Faculty of Mathematics and Computer Science, University of Lodz, Poland

^*Corresponding author: Wiktor Burakowski
^*Corresponding author: Wiktor Burakowski

Dedicated to Professor Avner Friedman in celebration of his 90th birthday.

Received: October 2024

Revised: February 2025

Early access: March 29, 2025

Published online: March 29, 2025

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Abstract

We discuss the class of nonlinear elliptic equations $ \Delta u(x) + f(x, u(x)) + g(x, u)x \cdot\nabla 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.

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Mathematics Subject Classification: 35J15, 35B09, 35B40.

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