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Maximal existence domains of positive solutions for two-parametric systems of elliptic equations. (English) Zbl 1342.35100
This paper is concerned with the elliptic system $$-\Delta_pu=\lambda |u|^{p-2}u+c_1f(x)|u|^{\alpha-2}|v|^\beta u$$ in $$\Omega$$, $$-\Delta_qv=\mu |v|^{q-2}v+c_2|u|^\alpha|v|^{\beta-2}v$$ in $$\Omega$$, $$u=v=0$$ on $$\partial\Omega$$. Here, $$\Omega$$ is a $$C^{1,\delta}$$ bounded domain in $$\mathbb R^n$$, $$n\geq 1$$, $$\lambda,\mu\in\mathbb R$$, $$c_1,c_2>0$$, $$p,q>1$$, $$\alpha,\beta\geq 1$$ and $$f\in L^\infty(\Omega)$$ is allowed to change sign.
The authors are concerned with the maximal existence domain of nonnegative solutions in the $$(\lambda,\mu)$$-plane. In this sense, two continuous and monotone curves $$\Gamma_f$$ and $$\Gamma_g$$ are emphasized which are the lower and upper bounds for the maximal domain of existence of weak positive solutions. The curve $$\Gamma_f$$ determines the maximal domain of the applicability of the Nehari manifold and fibering methods. The curve $$\Gamma_g$$ is obtained explicitly via minimax variational principle of the extended functional method.

MSC:
 35J50 Variational methods for elliptic systems 35J60 Nonlinear elliptic equations 35J70 Degenerate elliptic equations 35R05 PDEs with low regular coefficients and/or low regular data
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