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Positive solutions for semilinear Dirichlet problems in an annulus. (English) Zbl 0768.35029

This paper considers the existence of positive radial solutions of \(- \Delta u=g(| x|)f(u)\) in an annulus with zero boundary conditions. The main new idea here is to use the mountain pass lemma, which allows the proof of existence results in a simple, direct way when \(f\) is continuous. The case of discontinuous nonlinearities uses Chang’s version of the mountain pass lemma for nonsmooth functionals.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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[1] Ambrosetti, A.; Badaile, M., The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl., 140, No. 2, 363-373 (1989) · Zbl 0687.35033
[2] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[3] Ambrosetti, A.; Struwe, M., Appl. Math. Lett., 2, No. 2, 183-186 (1989) · Zbl 0709.35081
[4] Ambrosetti, A.; Turner, R. E.L, Some discontinuous variational problems, Differential Integral Equations, 1, No. 3, 341-349 (1988) · Zbl 0728.35037
[6] Bandle, C.; Coffman, C. V.; Marcus, M., Nonlinear elliptic problems in annular domains, J. Differential Equations, 69, 332-345 (1987) · Zbl 0618.35043
[7] Chang, K. C., Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80, 102-129 (1981) · Zbl 0487.49027
[8] de Figuereido, D.; Lions, P. L.; Nussbaum, R. D., A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61, 41-63 (1982) · Zbl 0452.35030
[9] Garaizar, X., Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations, 70, 69-92 (1987) · Zbl 0651.35033
[10] Gidas, B.; Ni, W.-M; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020
[11] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6, No. 3, 883-901 (1981) · Zbl 0462.35041
[12] Lin, S.-S, On the existence of positive radial solutions for nonlinear elliptic equations in annular domains, J. Differential Equations, 81, 221-233 (1989) · Zbl 0691.35036
[13] Morrey, C. B., Multiple Integrals in Calculus of Variations (1966), Springer: Springer Berlin · Zbl 0142.38701
[14] Stuart, C. A., Differential equations with discontinuous nonlinearities, Arch. Rational Mech. Analysis, 63, 59-75 (1976) · Zbl 0393.34010
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