Existence of Weak Solutions for Nonlinear Systems Involving Degenerated p-Laplacian Operators
DOI:
https://doi.org/10.5644/SJM.05.1.04Keywords:
Weak solution, nonlinear elliptic system, p-Laplacian, monotone operatorsAbstract
We study the existence of weak solutions for the nonlinear system
\begin{equation*}
\left.
\begin{aligned}
-\Delta _{P,_{p}}u&=a(x)|u|^{p-2}u-b(x)|u|^{\alpha }|v|^{\beta }v+f, \\
-\Delta _{Q,q}v&=-c(x)|u|^{\alpha }|v|^{\beta }u+d(x)|v|^{q-2}v+g,
\end{aligned}
\right\}
\end{equation*}
where, the degenerated p-Laplacian is defined as $\Delta _{P,p}u=div [P(x)$ $|\nabla u|^{p-2}\nabla u].$ We prove the existence of weak solutions for this system defined on bounded domains using the theory of monotone operators. We also consider the case of an unbounded domain.
2000 Mathematics Subject Classification. 35J67, 35J55, 47H07
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