Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition

Repovš, Dušan (author)

We study the degenerate elliptic equation ▫$$ -\operatorname{div}(|x|^\alpha \nabla u) = f(u) + t\phi(x) + h(x)$$▫ in a bounded open set ▫$\Omega$▫ with homogeneous Neumann boundary condition, where ▫$\alpha \in (0,2)$▫ and ▫$f$▫ has a linear growth. The main result establishes the existence of real numbers and ▫$t^\ast$▫ such that the problem has at least two solutions if ▫$t \leq t_\ast$▫, there is at least one solution if ▫$t_\ast < t \leq t^\ast$▫, and no solution exists for all ▫$t > t^\ast$▫. The proof combines a priori estimates with topological degree arguments.

• Language: English
• Keywords: Ambrosetti-Prodi problem, degenerate potential, topological degree, anisotropic continuous media
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2018
• Number of pages: art. no. 41, str. 1-10
• Numbering: Vol. 2018
• PID: 20.500.12556/RUL-109446
• UDC: 517.956
• ISSN on article: 1072-6691
• COBISS.SI-ID: 18249305
• Publication date in RUL: 03.09.2019

Record is a part of a journal

• Title: Electronic journal of differential equations
• Shortened title: Electr. j. differ. equ.
• Publisher: Southwest Texas State University, University of North Texas
• ISSN: 1072-6691
• COBISS.SI-ID: 7027289