Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball. (English) Zbl 0656.35048
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.
MSC:
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
radially symmetric superlinear Dirichlet problem; infinitely many solutions; rapidly growing nonlinearities; critical exponent; Sobolev embedding theorem; energy analysis; phase-plane angle analysis; singular ordinary differential equationReferences:
| [1] | Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030 |
| [2] | Abbas Bahri and Henri Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1 – 32. · Zbl 0476.35030 |
| [3] | 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). · Zbl 0589.58007 |
| [4] | 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). · Zbl 0487.34037 |
| [5] | 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. · Zbl 0695.34018 · doi:10.1016/0022-247X(88)90420-9 |
| [6] | Maria J. Esteban, Multiple solutions of semilinear elliptic problems in a ball, J. Differential Equations 57 (1985), no. 1, 112 – 137. · Zbl 0519.35031 · doi:10.1016/0022-0396(85)90073-7 |
| [7] | Svatopluk Fučík and Vladimír Lovicar, Periodic solutions of the equation \?^{\(^{\prime}\)\(^{\prime}\)}(\?)+\?(\?(\?))=\?(\?), Časopis Pěst. Mat. 100 (1975), no. 2, 160 – 175. |
| [8] | Paul H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), no. 2, 753 – 769. · Zbl 0589.35004 |
| [9] | 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. · Zbl 0456.35031 · doi:10.1007/BF01299609 |
| [10] | Michael Struwe, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math. (Basel) 39 (1982), no. 3, 233 – 240. · Zbl 0496.35034 · doi:10.1007/BF01899529 |
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.
