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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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