Existence of positive solutions to nonlinear elliptic systems involving gradient term and reaction potential
In this note we study the elliptic system $$\displaylines{ -\Delta u = z^p+f(x) \quad \text{in }\Omega , \cr -\Delta z = |\nabla u|^{q}+g(x) \quad \text{in }\Omega , \cr z,u > 0 \quad \text{in }\Omega ,\cr z=u= 0 \quad \text{on }\partial \Omega, }$$ where $\Omega \subset \mathbb{R}^{N}$...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2017-04-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2017/113/abstr.html |
| Summary: | In this note we study the elliptic system
$$\displaylines{
-\Delta u = z^p+f(x) \quad \text{in }\Omega , \cr
-\Delta z = |\nabla u|^{q}+g(x) \quad \text{in }\Omega , \cr
z,u > 0 \quad \text{in }\Omega ,\cr
z=u= 0 \quad \text{on }\partial \Omega,
}$$
where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain, p>0, $0<q\le 2$
with pq<1 and f,g are two nonnegative measurable functions.
The main result of this work is to analyze the interaction between the potential
and the gradient terms in order to get the existence of a positive solution. |
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| ISSN: | 1072-6691 |