×

On the principal eigenvalue of degenerate quasilinear elliptic systems. (English) Zbl 1157.35077

Some properties of the principle eigenvalue of a degenerate quasilinear system are studied in the paper. It is proved that this eigenvalue is simple, unique up to positive eigenfunctions and isolated. The regularity of corresponding eigenfunctions is established under certain restrictions on the given data. The extension of the main result to unbounded domains is also discussed.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J70 Degenerate elliptic equations
35J50 Variational methods for elliptic systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allegretto, Sturm theorems for degenerate elliptic equations, Proc. Amer. Math. Soc. 129 pp 165– (2001) · Zbl 0982.35007
[2] Allegretto, A Picone’s identity for the p -Laplacian and applications, Nonlinear Anal. 32 pp 819– (1998)
[3] Anane, Simplicitè et isolation de la premiére valuer propre du p -Laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 pp 725– (1987)
[4] Anane, First order spectrum for elliptic system and nonresonance problem, Numer. Algorithms 21 pp 9– (1999) · Zbl 0956.35094
[5] Arapostathis, Harnack’s inequality for cooperative weakly coupled elliptic systems, Comm. Partial Differential Equations 24 pp 1555– (1999) · Zbl 0934.35039
[6] Bensoussan, Nonlinear systems of elliptic equations with natural growth conditions and sign conditions, Appl. Math. Optim. 46 pp 143– (2002) · Zbl 1077.35046
[7] A. Bensoussan, and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications (Springer, Berlin, 2002). · Zbl 1055.35002
[8] M. S. Berger, Nonlinearity and Functional Analysis (Academic Press, New York, 1977). · Zbl 0368.47001
[9] Birindelli, A class of quasilinear elliptic systems arising in image segmentation, NoDEA Nonlinear Differential Equations Appl. 5 pp 445– (1998)
[10] Boccardo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 9 pp 309– (2002)
[11] Bozhkov, Existence of multiple solutions for quasilinear systems via Fibering method, J. Differential Equations 190 pp 239– (2003) · Zbl 1021.35034
[12] Brown, On a system of reaction-diffusion equations describing a population with two age groups, J. Math. Anal. Appl. 282 pp 444– (2003) · Zbl 1031.35057
[13] Choi, Global existence of solutions of a strongly coupled quasilinear parabolic system with applications to electrochemistry, J. Differential Equations 194 pp 406– (2003) · Zbl 1036.35073
[14] Chuong, Existence results for a semilinear parametric problem with Grushin type operator, Electron. J. Differ. Equ. 107 pp 1– (2005) · Zbl 1170.35395
[15] Constantin, Global solutions for quasilinear parabolic systems, J. Differential Equations 197 pp 73– (2004) · Zbl 1043.35066
[16] Dancer, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal. 34 pp 292– (2002) · Zbl 1055.35046
[17] R. Dautray, and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. I: Physical Origins and Classical Methods (Springer-Verlag, Berlin, 1985). · Zbl 0766.47001
[18] Djellit, Existence of solutions for a class of elliptic systems in \(\mathbb{R}\)N involving the p -Laplacian, Electron. J. Differ. Equ. 56 pp 1– (2003) · Zbl 1109.35322
[19] Drábek, Bifurcation problems for the p -Laplacian in \(\mathbb{R}\)N, Trans. Amer. Math. Soc. 349 pp 171– (1997) · Zbl 0868.35010
[20] P. Drábek, A. Kufner, and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities (Walter de Gruyter & Co., Berlin, 1997). · Zbl 0894.35002
[21] Drábek, Existence and regularity of solutions for nonlinear elliptic systems in \(\mathbb{R}\)N, Atti Sem. Mat. Fis. Univ. Modena 50 pp 161– (2002)
[22] Drábek, Multiple nonsemitrivial solutions for quasilinear elliptic systems, Differential Integral Equations 16 pp 1519– (2003) · Zbl 1073.35025
[23] Fleckinger, Principal eigenvalues for some quasilinear elliptic equations on \(\mathbb{R}\)N, Adv. Differential Equations 2 pp 981– (1997)
[24] Kandilakis, A subsolution-supersolution method for quasilinear systems, Electron. J. Differ. Equ. 97 pp 1– (2005) · Zbl 1210.35023
[25] Kandilakis, The first eigenvalue of p -Laplacian systems with nonlinear boundary conditions, Boundary Value Problems 2005 (3) pp 307– (2005) · Zbl 1109.35082
[26] Karachalios, Convergence towards attractors for a degenerate Ginzburg-Landau equation, Z. Angew. Math. Phys. 56 pp 11– (2005) · Zbl 1181.35133
[27] Karachalios, On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. and Partial Differ. Equ. 25 (3) pp 361– (2006)
[28] Kristaly, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains, Proc. Edinb. Math. Soc., II. Ser. 48 pp 465– (2005)
[29] Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations 197 pp 293– (2004) · Zbl 1210.35122
[30] Li, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math. LVI pp 892– (2003) · Zbl 1125.35339
[31] Lindqvist, On the equation div(|u |p -2u) + {\(\lambda\)} |u |p -2u = 0, Proc. Amer. Math. Soc. 109 pp 157– (1990) · Zbl 0714.35029
[32] Manasevich, The spectrum of p -Laplacian systems with various boundary conditions and applications, Adv. Differential Equations 5 pp 1289– (2000)
[33] El Manouni, On some nonlinear elliptic systems with coercive perturbations in \(\mathbb{R}\)N, Rev. Mat. Complut. 16 (2) pp 483– (2003) · Zbl 1086.35032
[34] Martinez, Isolation and simplicity for the first eigenvalue of the p -Laplacian with a nonlinear boundary condition, Abstr. Appl. Anal. 7 (5) pp 287– (2002)
[35] Maz’ya, Solutions for quasilinear nonsmooth evolution systems in Lp, Arch. Ration. Mech. Anal. 171 pp 219– (2004)
[36] J. D. Murray, Mathematical Biology, II: Spatial Models and Biomedical Applications (Springer-Verlag, New York - Berlin - Heidelberg, 2003). · Zbl 1006.92002
[37] De Nápoli, Quasilinear elliptic systems of resonant type and nonlinear eigenvalue problems, Abstr. Appl. Anal. 7 (3) pp 155– (2002) · Zbl 1005.35036
[38] De Nápoli, Estimates for eigenvalues of quasilinear elliptic systems, J. Differential Equations 227 pp 102– (2006) · Zbl 1100.35077
[39] Del Pezzo, An optimization problem for the first eigenvalue of the p -Laplacian plus a potential, Comm. Pure Appl. Anal. 5 (4) pp 675– (2006) · Zbl 1175.35092
[40] Poulou, Global bifurcation results on degenerate quasilinear elliptic systems on \(\mathbb{R}\)N, Nonlinear Anal., Theory Methods Appl. 66 pp 214– (2007)
[41] Serag, Maximum principle and existence of positive solutions for nonlinear systems on \(\mathbb{R}\)N, Electron. J. Differ. Equ. 85 pp 1– (2005) · Zbl 1140.35366
[42] Stavrakakis, Bifurcation results for some quasilinear elliptic systems on \(\mathbb{R}\)N, Adv. Differential Equations 8 pp 315– (2003) · Zbl 1229.35068
[43] de Thélin, Premiere valeur propre d’un systeme elliptique non lineaire, Rev. Mat. Apl. 13 pp 1– (1992) · Zbl 0779.35081
[44] Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 pp 191– (1984) · Zbl 0561.35003
[45] Zhang, Remarks on a class of quasilinear elliptic systems involving the (p, q)-Laplacian, Electron. J. Differ. Equ. 20 pp 1– (2005)
[46] Zographopoulos, On a class of degenerate potential elliptic system, NoDEA, Nonlinear Differ. Equ. Appl. 11 pp 191– (2004) · Zbl 1073.35080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.