Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition
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...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2018-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/41/abstr.html |
| Summary: | 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. |
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| ISSN: | 1072-6691 |