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A strong maximum principle for some quasilinear elliptic equations. (English) Zbl 0561.35003

The main result is the following maximum principle: if \(\beta\) is a real- valued, non-decreasing function of a real variable with \(\beta (0)=0\), and f is non-negative almost everywhere on \(\Omega \subset {\mathbb{R}}^ n\), then every non-negative weak solution of \(-\Delta u+\beta (u)=f\) is positive everywhere if and only if \(\int (\beta (s)s)^{-1/2}ds\) diverges at \(s=0+\). The result extends to certain quasi-linear equations.
Reviewer: J.F.Toland

MSC:

35B50 Maximum principles in context of PDEs
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
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[1] Aris R (1975) The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon Press, Oxford · Zbl 0315.76052
[2] Bandle C, Sperb RP, Stakgold I (in press) Diffusion-reaction with monotone kynetics. J Nonlinear Analysis
[3] Bénilan P (1978) Opérateurs accrétifs et semigroupes dansL p (1?p??). In: Fujita H (ed) Japan-France Seminar 1976. Japan Society for the Promotion of Science: Tokyo
[4] Bénilan P, Brézis H, Crandall MG (1975) A semilinear equation inL 1(? n ). Ann Scuola Norm Sup Pisa 4:523-555
[5] Bertsch M, Kersner R, Peletier LA (in press) Positivity versus localization in degenerate diffusion equations · Zbl 0596.35073
[6] Brézis H, Véron L (1980) Removable singularities of some nonlinear elliptic equations. Arch Rat Mech Anal75 1-6 · Zbl 0459.35032
[7] Díaz JI, Hernández J (to appear) On the existence of a free boundary for a class of reactiondiffusion systems. Madison Res Center TS Report 2330. SIAM J Math Anal
[8] Díaz JI, Herrero MA (1981) Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems. Proc Royal Soc Ed 89A:249-258 · Zbl 0478.35083
[9] di Benedetto E (1983)C 1+? local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis 7:827-850 · Zbl 0539.35027
[10] Friedman A, Phillips D (in press) The free boundary of a semilinear elliptic equation. Tr Amer Math Soc · Zbl 0552.35079
[11] Gilbarg D, Trudinger NS (1977) Elliptic Differential Equations of Second Order. Springer Verlag, Berlin · Zbl 0361.35003
[12] Hopf E (1927) Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitz Ber Preuss Akad Wissensch, Berlin. Math Phys kl 19 · JFM 53.0454.02
[13] Kato T (1972) Schrödinger operators with singular potentials. Israel J Math 13:135-148 · Zbl 0246.35025
[14] Tolksdorf P (1984) Regularity for a more general class of quasilinear elliptic equations. J Diff Equations 51:126-150 · Zbl 0522.35018
[15] Vázquez JL, Véron L (in press) Isolated singularities of some semilinear elliptic equations. J Diff Eq
[16] Vázquez JL, Véron L (in press) Singularities of elliptic equations with an exponential nonlinearity. Mathematische Annalen
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