Existence of multiple solutions for a semilinear problem and a counterexample by E. N. Dancer

Alfonso Castro; Guillermo Reyes

doi:10.3934/cpaa.2017055

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2017, Volume 16, Issue 4: 1135-1146. Doi: 10.3934/cpaa.2017055

This issue Previous Article Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains Next Article Nonlinear Dirichlet problems with double resonance

Existence of multiple solutions for a semilinear problem and a counterexample by E. N. Dancer

Alfonso Castro^1, and
Guillermo Reyes^2, ,

1.
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
2.
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

^* Corresponding author: G. Reyes
^* Corresponding author: G. Reyes

Received: December 31, 2015

Revised: January 31, 2017

Published: May 2017

This work was partially supported by a grant from the Simons Foundations (# 245966 to Alfonso Castro). It was completed as part of Alfonso Castrós Cátedra de Excelencia at the Universidad Complutense de Madrid funded by the Consejería de Educaci′on, Juventud y Deporte de la Comunidad de Madrid

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Abstract

As shown by Dancer's counterexample ^[12], one cannot expect to have (generically) more than five solutions to the semilinear boundary value problem

$\mathbf{(P)}\qquad \left\{\begin{array}{ ll}-\Delta u =f(u, x)\quad&\text{in}B:=\{x\in\mathbb{R}^n:\, |x|<1\}, \\u =0&\text{on}\partial B\end{array}\right.$

when $f(0, x)=0$ and $\partial f/\partial u$ crosses the first two eigenvalues of $-\Delta$ with Dirichlet boundary conditions on $\partial B$, as $|u|$ grows from $0$ to $\infty$. Despite the fact that the nonlinearity $ f$ in ^[12] can be taken "arbitrarily close" to autonomous, we prove that the dependence of $f$ on $x$ is indeed essential in the arguments. More precisely, we show that (P) with $f$ independent of $x$ and satisfying the assumptions in ^[12] has at least six, and generically seven solutions. Under more stringent conditions on the non-linearity, we prove that there are up to nine solutions. Importantly, we do not assume any symmetry on $f$ for any of our results.

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Mathematics Subject Classification: 35J20, 35J25, 35J61, 35B38.

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