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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.

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
Full Text: DOI arXiv
[1] Díaz JI, Nonlinear partial differential equations and free boundaries (1985)
[2] Yin H-M, Q. Appl. Math pp 47– (2001) · Zbl 1030.35109
[3] DOI: 10.1007/BF01206962 · Zbl 0809.35022
[4] DOI: 10.1070/IM2002v066n06ABEH000408 · Zbl 1112.35311
[5] DOI: 10.1512/iumj.1991.40.40049 · Zbl 0773.35020
[6] DOI: 10.1016/S0022-0396(02)00112-2 · Zbl 1021.35034
[7] DOI: 10.1016/j.camwa.2007.03.025 · Zbl 1155.35025
[8] DOI: 10.1016/j.na.2012.06.007 · Zbl 1259.35080
[9] DOI: 10.1186/1687-1847-2013-1 · Zbl 1365.05013
[10] DOI: 10.1017/S0308210512001564 · Zbl 1302.35168
[11] DOI: 10.1007/s10688-007-0002-2 · Zbl 1124.35307
[12] Ivanov AA, Zh. Vychisl. Mat. Mat. Fiz 53 pp 350– (2013)
[13] DOI: 10.1080/01630569708816767 · Zbl 0884.65103
[14] DOI: 10.1016/j.physd.2007.10.007 · Zbl 1145.35405
[15] DOI: 10.1080/17476933.2011.575461 · Zbl 1229.35052
[16] Anane A. Etude des valeurs propres et de la résonance pour l’opérateurp-Laplacien [Study of the eigenvalues and the resonance for the operator p-Laplacian] [PhD thesis]. Bruxelles: Thése de doctorat, ULB; 1987.
[17] DOI: 10.1512/iumj.1972.21.21079 · Zbl 0223.35038
[18] DOI: 10.1515/9783110804775
[19] DOI: 10.1016/0362-546X(88)90053-3 · Zbl 0675.35042
[20] DOI: 10.1007/BF01449041 · Zbl 0561.35003
[21] DOI: 10.1016/S0362-546X(97)00530-0 · Zbl 0930.35053
[22] DOI: 10.1016/j.mcm.2010.02.004 · Zbl 1201.35102
[23] DOI: 10.1007/s11401-013-0772-1 · Zbl 1278.35068
[24] Kinderlehrer D, An introduction to variational inequalities and their applications 31 (1980)
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