Original Paper
On the principal eigenvalue of degenerate quasilinear elliptic systems
Nikolaos B. Zographopoulos,
Corresponding Author
Nikolaos B. Zographopoulos
Department of Science, Division of Mathematics, Technical University of Crete, 73100 Chania, Greece
Phone: +30 28210 37756, Fax: +30 28210 37842Search for more papers by this authorNikolaos B. Zographopoulos,
Corresponding Author
Nikolaos B. Zographopoulos
Department of Science, Division of Mathematics, Technical University of Crete, 73100 Chania, Greece
Phone: +30 28210 37756, Fax: +30 28210 37842Search for more papers by this authorAbstract
We study the properties of the positive principal eigenvalue of a degenerate quasilinear elliptic system. We prove that this eigenvalue is simple, unique up to positive eigenfunctions and isolated. Under certain restrictions on the given data, the regularity of the corresponding eigenfunctions is established. The extension of the main result in the case of an unbounded domain is also discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
References
- [1] W. Allegretto, Sturm theorems for degenerate elliptic equations, Proc. Amer. Math. Soc. 129, 165–174 (2001).
- [2] W. Allegretto, and Y. X. Huang, A Picone's identity for the p -Laplacian and applications, Nonlinear Anal. 32, 819–830 (1998).
- [3] A. Anane, Simplicitè et isolation de la premiére valuer propre du p -Laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305, 725–728 (1987).
- [4] A. Anane, O. Chakrone, and Z. El Allali, First order spectrum for elliptic system and nonresonance problem, Numer. Algorithms 21, 9–21 (1999).
- [5] A. Arapostathis, M. K. Ghosh, and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems, Comm. Partial Differential Equations 24, 1555–1571 (1999).
- [6] A. Bensoussan, and L. Boccardo, Nonlinear systems of elliptic equations with natural growth conditions and sign conditions, Appl. Math. Optim. 46, 143–166 (2002).
- [7] A. Bensoussan, and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications (Springer, Berlin, 2002).
- [8] M. S. Berger, Nonlinearity and Functional Analysis (Academic Press, New York, 1977).
- [9] I. Birindelli, and S. Finzi Vita, A class of quasilinear elliptic systems arising in image segmentation, NoDEA Nonlinear Differential Equations Appl. 5, 445–459 (1998).
- [10] L. Boccardo, and D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 9, 309–323 (2002).
- [11] Y. Bozhkov, and E. Mitidieri, Existence of multiple solutions for quasilinear systems via Fibering method, J. Differential Equations 190, 239–267 (2003).
- [12] K. J. Brown, and Y. Zhang, On a system of reaction-diffusion equations describing a population with two age groups, J. Math. Anal. Appl. 282, 444–452 (2003).
- [13] Y. S. Choi, Z. Huan, and R. Lui, Global existence of solutions of a strongly coupled quasilinear parabolic system with applications to electrochemistry, J. Differential Equations 194, 406–432 (2003).
- [14] N. M. Chuong, and T. D. Ke, Existence results for a semilinear parametric problem with Grushin type operator, Electron. J. Differ. Equ. 107, 1–12 (2005).
- [15] A. Constantin, J. Escher, and Z. Yin, Global solutions for quasilinear parabolic systems, J. Differential Equations 197, 73–84 (2004).
- [16] E. N. Dancer, and Y. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal. 34, 292–314 (2002).
- [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).
- [18] A. Djellit, and A. Tas, Existence of solutions for a class of elliptic systems in ℝN involving the p -Laplacian, Electron. J. Differ. Equ. 56, 1–8 (2003).
- [19] P. Drábek, and Y. X. Huang, Bifurcation problems for the p -Laplacian in ℝN , Trans. Amer. Math. Soc. 349, 171–188 (1997).
- [20] P. Drábek, A. Kufner, and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities (Walter de Gruyter & Co., Berlin, 1997).
- [21] P. Drábek, S. El Manoumi, and A. Touzani, Existence and regularity of solutions for nonlinear elliptic systems in ℝN , Atti Sem. Mat. Fis. Univ. Modena 50, 161–172 (2002).
- [22] P. Drábek, N. M. Stavrakakis, and N. B. Zographopoulos, Multiple nonsemitrivial solutions for quasilinear elliptic systems, Differential Integral Equations 16, 1519–1531 (2003).
- [23] J. Fleckinger, R. Manasevich, N. M. Stavrakakis, and F. de Thelin, Principal eigenvalues for some quasilinear elliptic equations on ℝN , Adv. Differential Equations 2, 981–1003 (1997).
- [24] D. A. Kandilakis, and M. Magiropoulos, A subsolution-supersolution method for quasilinear systems, Electron. J. Differ. Equ. 97, 1–5 (2005).
- [25] D. A. Kandilakis, M. Magiropoulos, and N. B. Zographopoulos, The first eigenvalue of p -Laplacian systems with nonlinear boundary conditions, Boundary Value Problems 2005, No. 3, 307–321 (2005).
- [26] N. I. Karachalios, and N. B. Zographopoulos, Convergence towards attractors for a degenerate Ginzburg–Landau equation, Z. Angew. Math. Phys. 56, 11–30 (2005).
- [27] N. I. Karachalios, and N. B. Zographopoulos, On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. and Partial Differ. Equ. 25, No. 3, 361–393 (2006).
- [28] A. 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, 465–477 (2005).
- [29] K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations 197, 293–314 (2004).
- [30] Y. Li, and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math. LVI, 892–925 (2003).
- [31] P. Lindqvist, On the equation div(|∇u |p –2∇u) + λ |u |p –2u = 0, Proc. Amer. Math. Soc. 109, 157–164 (1990).
- [32] R. Manasevich, and J. Mawhin, The spectrum of p -Laplacian systems with various boundary conditions and applications, Adv. Differential Equations 5, 1289–1318 (2000).
- [33]
S. El Manouni, and A. Touzani, On some nonlinear elliptic systems with coercive perturbations in ℝN , Rev. Mat. Complut. 16, No. 2, 483–494 (2003).
10.5209/rev_REMA.2003.v16.n2.16824 Google Scholar
- [34]
S. Martinez, and J. D. Rossi, Isolation and simplicity for the first eigenvalue of the p -Laplacian with a nonlinear boundary condition, Abstr. Appl. Anal. 7, No. 5, 287–293 (2002).
10.1155/S108533750200088X Google Scholar
- [35] V. Maz'ya, J. Elschner, J. Rehberg, and G. Schmidt, Solutions for quasilinear nonsmooth evolution systems in Lp , Arch. Ration. Mech. Anal. 171, 219–262 (2004).
- [36] J. D. Murray, Mathematical Biology, II: Spatial Models and Biomedical Applications (Springer-Verlag, New York – Berlin – Heidelberg, 2003).
- [37]
P. L. De Nápoli, and C. Marianni, Quasilinear elliptic systems of resonant type and nonlinear eigenvalue problems, Abstr. Appl. Anal. 7, No. 3, 155–167 (2002).
10.1155/S1085337502000829 Google Scholar
- [38] P. L. De Nápoli, and J. P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, J. Differential Equations 227, 102–115 (2006).
- [39] L. M. Del Pezzo, and J. F. Bonder, An optimization problem for the first eigenvalue of the p -Laplacian plus a potential, Comm. Pure Appl. Anal. 5, No. 4, 675–690 (2006).
- [40] M. N. Poulou, N. M. Stavrakakis, and N. B. Zographopoulos, Global bifurcation results on degenerate quasilinear elliptic systems on ℝN , Nonlinear Anal., Theory Methods Appl. 66, 214–227 (2007).
- [41] H. M. Serag, and E. A. El-Zahrani, Maximum principle and existence of positive solutions for nonlinear systems on ℝN , Electron. J. Differ. Equ. 85, 1–12 (2005).
- [42] N. M. Stavrakakis, and N. B. Zographopoulos, Bifurcation results for some quasilinear elliptic systems on ℝN , Adv. Differential Equations 8, 315–336 (2003).
- [43] F. de Thélin, Premiere valeur propre d'un systeme elliptique non lineaire, Rev. Mat. Apl. 13, 1–8 (1992).
- [44] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12, 191–202 (1984).
- [45] G. Zhang, X. Liu, and S. Liu, Remarks on a class of quasilinear elliptic systems involving the (p, q)-Laplacian, Electron. J. Differ. Equ. 20, 1–10 (2005).
- [46] N. B. Zographopoulos, On a class of degenerate potential elliptic system, NoDEA, Nonlinear Differ. Equ. Appl. 11, 191–199 (2004).
