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


35B50 Maximum principles in context of PDEs
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
Full Text: DOI


[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 · doi:10.1016/0362-546X(83)90061-5
[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
[13] Kato T (1972) Schrödinger operators with singular potentials. Israel J Math 13:135-148 · Zbl 0246.35025 · doi:10.1007/BF02760233
[14] Tolksdorf P (1984) Regularity for a more general class of quasilinear elliptic equations. J Diff Equations 51:126-150 · doi:10.1016/0022-0396(84)90105-0
[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
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.