×

Classical and weak solutions of a singular semilinear elliptic problem. (English) Zbl 0880.35043

Authors’ summary: The singular semilinear elliptic equation \(\Delta u+p(x)f(u)= 0\) is shown to have a unique positive classical solution in \(\mathbb{R}^n\) that decays to zero at \(\infty\) if \(p(x)\) is simply a nontrivial nonnegative continuous function satisfying \(\int^\infty_0 t(\max_{|x|=t}p(x))dt<\infty\), provided \(f\) is a nonincreasing continuously differentiable function on \((0,\infty)\). It is also shown that the equation has a unique weak \(H^1_0\)-solution on a bounded domain provided \(\int^\varepsilon_0 f(s)ds<\infty\) and \(p(x)\in L^2\).

MSC:

35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Real analysis lecture notes, (1982), Beijing UniversityMath. Dept
[2] Crandall, M.G.; Rabinowitz, P.H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. partial differential equations, 2, 193-222, (1977) · Zbl 0362.35031
[3] Edelson, A., Entire solutions of singular elliptic equations, J. math. anal. appl., 139, 523-532, (1989) · Zbl 0679.35003
[4] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice Hall Englewood Cliffs · Zbl 0144.34903
[5] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry and related problems via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[6] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1977), Springer-Verlag New York · Zbl 0691.35001
[7] Graham-Eagle, J., A variational approach to upper and lower solutions, IMA J. appl. math., 44, 181-184, (1990) · Zbl 0705.35047
[8] Kusano, T.; Swanson, C.A., Entire positive solutions of singular semilinear elliptic equations, Japan J. math., 11, 145-155, (1985) · Zbl 0585.35034
[9] Lair, A.V.; Shaker, A.W., Entire solution of a singular semilinear elliptic problem, J. math. anal. appl., 200, 498-505, (1996) · Zbl 0860.35030
[10] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. amer. math. soc., 111, 721-730, (1991) · Zbl 0727.35057
[11] Ni, W.M., On the elliptic equation δukxu(nn=0, its generalization and application in geometry, Indiana univ. math. J., 31, 493-529, (1982)
[12] del Pino, M.A., A global estimate for the gradient in a singular elliptic boundary value problem, Proc. roy. soc. Edinburgh sect. A, 122, 341-352, (1992) · Zbl 0791.35046
[13] Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana univ. math. J., 21, 979-1000, (1973) · Zbl 0223.35038
[14] Shaker, A.W., On singular semilinear elliptic equations, J. math. anal. appl., 173, 222-228, (1993) · Zbl 0785.35032
[15] Struwe, M., Variational methods, (1990), Springer-Verlag New York/Berlin
[16] Yosida, K., Functional analysis, (1980), Springer-Verlag New York/Berlin · Zbl 0217.16001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.