Singular limits solution for 2-dimensional elliptic problems involving exponential nonlinearities with sub-quadratic convection nonlinear gradient terms and singular weights
Given a bounded open regular set Ω of ℝ2$\mathbb {R}^2$, q1,...,qK∈Ω${q_1, \ldots , q_K \hspace*{-0.85358pt}\in \hspace*{-0.85358pt} \Omega }$, a regular bounded function ϱ:Ω→[0,+∞)${\varrho \hspace*{-0.56905pt}:\hspace*{-0.56905pt} \Omega \hspace*{-0.85358pt}\rightarrow \hspace*{-0.85358pt} [0,+\in...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2014-09-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2014-0004 |
| Summary: | Given a bounded open regular set Ω of ℝ2$\mathbb {R}^2$,
q1,...,qK∈Ω${q_1, \ldots , q_K \hspace*{-0.85358pt}\in \hspace*{-0.85358pt} \Omega }$, a regular
bounded function ϱ:Ω→[0,+∞)${\varrho \hspace*{-0.56905pt}:\hspace*{-0.56905pt} \Omega \hspace*{-0.85358pt}\rightarrow \hspace*{-0.85358pt} [0,+\infty )}$
and a bounded function V:Ω→[0,+∞)${V: \Omega \rightarrow [0,+\infty )}$, we give a sufficient
condition for the model problem
-Δu-λϱ(x)|∇u|q=ε2V(x)eu$
-\Delta u -\lambda \varrho (x)\vert \nabla u\vert ^q = \varepsilon ^2 V(x) e^u
$
to have a positive weak solution in Ω with u = 0 on
∂Ω${\partial \Omega }$, which is singular at each qi as the parameters ε and λ tend to 0, without considering any relation between them, essentially when the set of concentration points qi and the set of zeros of V are not necessarily disjoint and q∈[1,2)$q\in [1,2)$ is a real number. |
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| ISSN: | 2191-9496 2191-950X |