Previous |  Up |  Next

Article

Keywords:
Eigenvalue problem; Degenerate elliptic operator; Nonlinear systems; Positive solutions.
Summary:
In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems \[ \left\rbrace \begin{array}{ll}-\Delta _p u = f(x,u,v), &\quad \text{in} \ \Omega , -\Delta _p v = g(x,u,v), &\quad \text{in} \ \Omega , u = v = 0, &\quad \text{on} \ \partial \Omega , \end{array}\right.\] where $-\Delta _p$ is the $p$-Laplace operator, $p>1$ and $\Omega $ is a $C^{1,\alpha }$-domain in $\mathbb R^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)=\lambda _1 v^{p-1}$, $ g(x,u,v)=\lambda _2u^{p-1}$ and obtain a non-existence result of positive solutions for the general systems.
References:
[1] Y. Cheng: On the existence of radial solutions of a nonlinear elliptic BVP in an annulus. Math. Nachr. 165 (1994), 61–77. DOI 10.1002/mana.19941650106 | MR 1261363 | Zbl 0836.35050
[2] Y. Cheng: On the existence of radial solutions of a nonlinear elliptic equation on the unit ball. Nonlinear Analysis, TMA, 24:3 (1995), 287–307. MR 1312769 | Zbl 0819.35050
[3] Y. Cheng: An eigenvalue problem for a quasilinear elliptic equation. U.U.M.D. Report NO:19, 1994.
[4] Ph. Clément, D. de Figueiredo, E. Mitidieri: Positive solutions of semilinear elliptic systems. Comm. P.D.E. 17 (1992), 923–940. DOI 10.1080/03605309208820869 | MR 1177298
[5] Ph. Clément, R. Manásevich, E. Mitidieri: Positive solutions of quasilinear elliptic systems via blow up. Comm. P.D.E. 18 (1993), 2071-2116. MR 1249135
[6] R. Courant, D. Hilbert: Methods of Mathematical Physics II. Interscience, 1953. MR 0065391
[7] R. Dalmasso: Positive solutions of nonlinear elliptic systems. Annl. Pol. Math. LVIII.2 (1993), 201–213. MR 1239024 | Zbl 0791.35014
[8] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Second edition, Springer-Verlag, 1992.
[9] J. Heinonen, T. Kilpelaninen, O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford, University Press, 1993. MR 1207810
[10] H. A. Levine: The role of critical exponents in blow-up theorems. SIAM Review 32:2 (1990), 262–288. DOI 10.1137/1032046 | MR 1056055
[11] P. Lindqvist: On the equation div$(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2} u=0$. Proc. Amer. Math. Soci. 109 (1990), 157–164. DOI 10.1090/S0002-9939-1990-1007505-7 | MR 1007505 | Zbl 0714.35029
[12] P. Lions: On the existence of positive solutions of semilinear elliptic equations. SIAM Review 24 (1982), 441–467. DOI 10.1137/1024101 | MR 0678562 | Zbl 0511.35033
[13] M. Mitidieri: A Rellich type identity and applications. Comm. P.D.E. 18 (1993), 125–151. DOI 10.1080/03605309308820923 | MR 1211727 | Zbl 0816.35027
[14] L. Peletier, R.C.M. van der Vorst: Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equations. Diff Interg. Equa. 5 (1992), 747–767. MR 1167492
[15] S. Sakaguchi: Concavity property of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola. Norm. Pisa 14 (1987), 403–421. MR 0951227
[16] R.C.M. van der Vorst: Variational identities and applications to differential systems. Arch. rational Mech. Anal. 116c (1991), 375–398. MR 1132768 | Zbl 0796.35059
Partner of
EuDML logo