Electron. J. Differential Equations, Vol. 2018 (2018), No. 41, pp. 1-10.

Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition

Dusan D. Repovs

Abstract:
We study the degenerate elliptic equation
$$ -\hbox{div}(|x|^\alpha\nabla u) =f(u)+t\phi(x)+h(x) $$
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 $t_*$ and $t^*$ such that the problem has at least two solutions if $t\leq t_*$, there is at least one solution if $t_*<t\leq t^*$, and no solution exists for all $t>t^*$. The proof combines a priori estimates with topological degree arguments.

Submitted July 20, 2017. Published February 6, 2018.
Math Subject Classifications: 35J65, 35J25, 58E07.
Key Words: Ambrosetti-Prodi problem; degenerate potential; topological degree; anisotropic continuous media.

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Dusan D. Repovs
Faculty of Education and Faculty of Mathematics and Physics
University of Ljubljana
SI-1000 Ljubljana, Slovenia
email: dusan.repovs@guest.arnes.si

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