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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: January 26, 1999
MathSciNet review: 1458318
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Abstract: This paper is concerned with the multiplicity of radially symmetric solutions $u(x)$ to the Dirichlet problem

\begin{displaymath}\Delta u+f(u)=h(x)+c\phi(x)\end{displaymath}

on the unit ball $\Omega\subset\mathbf R^N$ with boundary condition $u=0$ on $\partial\Omega$. Here $\phi(x)$ is a positive function and $f(u)$ is a function that is superlinear (but of subcritical growth) for large positive $u$, while for large negative $u$ we have that $f'(u)<\mu$, where $\mu$ is the smallest positive eigenvalue for $\Delta\psi+\mu\psi=0$ in $\Omega$ with $\psi=0$ on $\partial\Omega$. It is shown that, given any integer $k\ge 0$, the value $c$ may be chosen so large that there are $2k+1$ solutions with $k$ or less interior nodes. Existence of positive solutions is excluded for large enough values of $c$.


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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: http://dx.doi.org/10.1090/S0002-9947-99-02110-8
PII: S 0002-9947(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