Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball

Castro, Alfonso; Kurepa, Alexandra

doi:10.1090/S0002-9939-1987-0897070-7

Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball
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by Alfonso Castro and Alexandra Kurepa PDF

Proc. Amer. Math. Soc. 101 (1987), 57-64 Request permission

Abstract:

In this paper we show that a radially symmetric superlinear Dirichlet problem in a ball has infinitely many solutions. This result is obtained even in cases of rapidly growing nonlinearities, that is, when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem. Our methods rely on the energy analysis and the phase-plane angle analysis of the solutions for the associated singular ordinary differential equation.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Abbas Bahri and Henri Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1–32. MR 621969, DOI 10.1090/S0002-9947-1981-0621969-9
Abbas Bahri and Pierre-Louis Lions, Remarques sur la théorie variationnelle des points critiques et applications, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 5, 145–147 (French, with English summary). MR 801948
Alfonso Castro and Alan C. Lazer, On periodic solutions of weakly coupled systems of differential equations, Boll. Un. Mat. Ital. B (5) 18 (1981), no. 3, 733–742 (English, with Italian summary). MR 641732
Alfonso Castro and R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlinearities. II, J. Math. Anal. Appl. 133 (1988), no. 2, 509–528. MR 954725, DOI 10.1016/0022-247X(88)90420-9
Maria J. Esteban, Multiple solutions of semilinear elliptic problems in a ball, J. Differential Equations 57 (1985), no. 1, 112–137. MR 788425, DOI 10.1016/0022-0396(85)90073-7
Svatopluk Fučík and Vladimír Lovicar, Periodic solutions of the equation $x^{^{\prime \prime }}(t)+g(x(t))=p(t)$, Časopis Pěst. Mat. 100 (1975), no. 2, 160–175. MR 0385239
Paul H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), no. 2, 753–769. MR 662065, DOI 10.1090/S0002-9947-1982-0662065-5
Michael Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. 32 (1980), no. 3-4, 335–364. MR 595426, DOI 10.1007/BF01299609
Michael Struwe, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math. (Basel) 39 (1982), no. 3, 233–240. MR 682450, DOI 10.1007/BF01899529