On the number of radially symmetric

solutions to Dirichlet problems with

jumping nonlinearities of superlinear order

Authors:
Alfonso Castro and Hendrik J. Kuiper

Journal:
Trans. Amer. Math. Soc. **351** (1999), 1919-1945

MSC (1991):
Primary 35J65, 34A10

DOI:
https://doi.org/10.1090/S0002-9947-99-02110-8

Published electronically:
January 26, 1999

MathSciNet review:
1458318

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the multiplicity of radially symmetric solutions to the Dirichlet problem

on the unit ball with boundary condition on . Here is a positive function and is a function that is superlinear (but of subcritical growth) for large positive , while for large negative we have that , where is the smallest positive eigenvalue for in with on . It is shown that, given any integer , the value may be chosen so large that there are solutions with or less interior nodes. Existence of positive solutions is excluded for large enough values of .

**1.**A. Ambrosetti and G. Prodi,*On the inversion of some differentiable mappings with singularities between Banach spaces*, Ann. Math. Pura Appl.**93**(1972), 231-246. MR**47:9377****2.**A. Castro and S. Gadam,*The Lazer McKenna conjecture for radial solutions in the ball*, Elec. J. Diff. Eq., 1993, No. 7, 1-6. MR**94j:35049****3.**A. Castro and A. Kurepa,*Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities*, Trans. Amer. Math. Soc.**315**(1989), 353-372. MR**90g:35053****4.**E. Hille,*Lectures on Ordinary Differential Equations*, Addison-Wesley, Reading, MA, 1969. MR**40:2939****5.**A. C. Lazer and P. J. McKenna,*Large amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis*, SIAM Review**32**(1990), 537-578. MR**92g:73059****6.**S.J. Poho\v{z}aev,*Eigenfunctions of the equation*, Soviet Math. Doklady**6**(1965), 1408-1411. MR**33:411**

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Additional Information

**Alfonso Castro**

Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203

**Hendrik J. Kuiper**

Affiliation:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287–1804

DOI:
https://doi.org/10.1090/S0002-9947-99-02110-8

Keywords:
Radially symmetric,
Dirichlet problem,
superlinear jumping nonlinearity,
nodal curves,
critical exponent.

Received by editor(s):
April 24, 1996

Published electronically:
January 26, 1999

Article copyright:
© Copyright 1999
American Mathematical Society