×

Remarks on sublinear elliptic equations. (English) Zbl 0593.35045

A nonlinear elliptic problem with positive solutions is considered: \[ - \Delta u=f(x,u)\quad in\quad D,\quad u>0,\quad u=0\quad on\quad \partial D,\quad D\quad bounded\quad domain\quad in\quad R^ n. \] The nonlinear term is sublinear and f(x,u)/u is decreasing on (0,\(\infty)\). Using these properties, the uniqueness is proved. The existence is established iff we have some inequalities concerning the eigenvalues of the associated operator [-\(\Delta\)-a(x)], where a(x) is given in terms of the limits of f(x,u)/u for \(u\to 0\) and \(u\to \infty\). The proof of the existence uses a minimization technique for the associated functional which is convex with respect to the variable \(u^ 2\).
Reviewer: G.Pasa

MSC:

35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Üniv. Math. J., 21, 125-146 (1971) · Zbl 0219.35037
[2] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044
[3] Benguria, R., The von Weizsäcker and exchange corrections in the Thomas-Fermi theory (1979), Princeton University, unpublished
[4] Benguria, R.; Brezis, H.; Lieb, E., The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Communs Math. Phys., 79, 167-180 (1981) · Zbl 0478.49035
[5] Berestycki, H., Le nombre de solutions de certains problémes semi-linéaires élliptiques, J. funct. Analysis, 40, 1-29 (1981) · Zbl 0452.35038
[6] Cohen, D.; Laetsch, T., Nonlinear boundary value problems suggested by chemical reactor theory, J. diff. Eqns, 7, 217-226 (1970) · Zbl 0201.43102
[7] De Figueiredo, D., Positive solutions of semilinear elliptic problems, Lecture Notes (1981), Sāo Paulo
[8] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin · Zbl 0361.35003
[9] Hess, P., On uniqueness of positive solutions of nonlinear elliptic boundary value problems, Math. Z., 154, 17-18 (1977) · Zbl 0352.35046
[10] Keller, H., Positive solutions of some nonlinear eigenvalue problems, J. Math. Mech., 19, 279-296 (1969) · Zbl 0188.17103
[11] Keller, H.; Cohen, D., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, 1361-1376 (1967) · Zbl 0152.10401
[12] Krasnoselskii, M., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen
[13] Laetsch, T., Uniqueness for sublinear boundary value problems, J. diff. Eqns, 13, 13-23 (1973) · Zbl 0247.35052
[14] Simpson, R. B.; Cohen, D., Positive solutions of nonlinear elliptic eigenvalue problems, J. Math. Mech., 19, 895-910 (1970) · Zbl 0195.39502
[15] Smoller, J.; Wasserman, A., Existence, uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations, Communs Math. Phys., 95, 129-159 (1984) · Zbl 0582.35046
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.