Positive solutions for singular elliptic problems depending on the gradient

Krzysztof Garbowski; Aleksandra Orpel

doi:10.3934/dcdsb.2025051

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2025, Volume 30, Issue 11: 4312-4323. Doi: 10.3934/dcdsb.2025051

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Positive solutions for singular elliptic problems depending on the gradient

Krzysztof Garbowski^{1, 2, ,} 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: Krzysztof Garbowski
^*Corresponding author: Krzysztof Garbowski

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

Received: October 2024

Revised: January 2025

Early access: March 31, 2025

Published online: March 31, 2025

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Abstract

We investigate the existence of asymptotically vanishing positive solutions of the following class of singular elliptic problems $ \Delta u(x)+f(x,u(x))-b(x)(u(x))^{-1}||\nabla u(x)||^{2}+g(x)x\cdot\nabla u(x) = 0\text{ in }\Omega_{R} = \left\{ x\in\mathbb{R}^{n},||x||>R>2\right\} $, $ n>2 $. Our approach is based on the sub-supersolution method for bounded sub-domains, as well as the unbounded domain approximation method combined with some classical convergence procedure. The rate of decay of the solutions is also described. In the first part of the paper we formulate some auxiliary results concerning solutions for the case when $ b\equiv0 $. These results play the crucial role in the proof of the existence of solution for our singular problem.

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Mathematics Subject Classification: 35J15, 35J75, 35B09.

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References

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