Doctoral School of Exact and Natural Science
University of Lodz, Poland
email: krzysztof.garbowski@wmii.uni.lodz.pl
Krzysztof Garbowski
Abstract:
In this article, we study the existence of classical, positive solutions that vanish asymptotically at infinity to
the class of singular elliptic problems with equation
$$
\Delta u(x)+f(x,u(x))-b(x)(u(x))^{-1}\|\nabla u(x)\|^2+g(x)x\cdot\nabla
u(x)=0
$$
in the exterior domain denoted by \(\Omega_{R}=\{x\in\mathbb{R}^n,\|x\| >R >1\}\),
\(n >2\). Moreover, we study the problem with additional boundary condition \(u(x) = S\) when
\(\|x\|=R\). In the first part of the paper, we present auxiliary results
concerning existence and multiplicity of solutions for the case when \(b\equiv0\).
These results play a crucial role in the proof of the existence of solution for our
singular problem in the second part of this paper.
We base our approach on the sub-super solution method for bounded sub-domains,
as well as the approximation method for the unbounded domain combined with some
classical convergence procedure.
Submitted July 13, 2026. Published October 15, 2026.
Math Subject Classifications: 35J15, 35J75, 35B09.
Key Words: Boundary condition; exterior domain; positive solutions;
singular elliptic problem; sub-super solution method.
DOI: 10.58997/ejde.2026.81
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Krzysztof Garbowski Doctoral School of Exact and Natural Science University of Lodz, Poland email: krzysztof.garbowski@wmii.uni.lodz.pl |
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