# American Institute of Mathematical Sciences

July  2017, 16(4): 1135-1146. doi: 10.3934/cpaa.2017055

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

 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

Received  January 2016 Revised  February 2017 Published  April 2017

Fund Project: 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

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.
Citation: Alfonso Castro, Guillermo Reyes. Existence of multiple solutions for a semilinear problem and a counterexample by E. N. Dancer. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1135-1146. doi: 10.3934/cpaa.2017055
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