    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 , 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  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  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
##### References:
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##### References:
  A. Aftalion and F. Pacella, Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C. R. Acad. Sci. Paris, Ser. Ⅰ, 339 (2004), 339-344.  doi: 10.1016/j.crma.2004.07.004.  Google Scholar  H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc., 85 (1982), 591-595.  doi: 10.2307/2044072.  Google Scholar  B. Breuer, P. J. McKenna and M. Plum, Multiple Solutions for a semilinear boundary value problem: a computational multiplicity proof, Journal of Differential Equations, 195 (2003), 243-269.  doi: 10.1016/S0022-0396(03)00186-4.  Google Scholar  A. Castro, Métodos de reducción via minimax, Primer Simposio Colombiano de Análisis Funcional, Medellín, Colombia, (1981). Google Scholar  A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal., 25 (1994), 1554-1561.  doi: 10.1137/S0036141092230106.  Google Scholar  A. Castro, J. Cossio and C. Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem without symmetries, Annali di Matematica Pura ed Applicata, 192 (2013), 607-619.  doi: 10.1007/s10231-011-0239-5.  Google Scholar  Castro, Alfonso, Drá bek, Pavel, Neuberger and M. John, A sign changing solution for a superlinear Dirichlet problem Ⅱ. Proceedings of the Fifth Mississippi State Conference on Differential Equations and Computational Simulations (Mississippi State, MS, 2001), 101-107 (electronic), Electron. J. Differ. Equ. Conf. , 10, Southwest Texas State Univ. , San Marcos, TX, 2003. Google Scholar  A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Annali di Matematica Pura ed Applicata, 120 (1979), 113-137.  doi: 10.1007/BF02411940.  Google Scholar  K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math., 120, 34 (1981), 693-712.  doi: 10.1002/cpa.3160340503.  Google Scholar  K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Springer Verlag, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar  D. Clark, A variant of the Lyusternik-Schnirelmann theory, Indiana U. Math. J., 22 (1973), 65-74.  doi: 10.1512/iumj.1972.22.22008.  Google Scholar  E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann., 272 (1985), 421-440.  doi: 10.1007/BF01455568.  Google Scholar  E. N. Dancer and Du Yihong, A note on multiple solutions of some semilinear elliptic problems, J. Math. Anal. Appl., 211 (1997), pp. 626-640.  doi: 10.1006/jmaa.1997.5471.  Google Scholar  H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509. Google Scholar  L. Hollman and P. J. McKenna, A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence, Commun. Pure Appl. Anal., 10 (2011), 785-802.  doi: 10.3934/cpaa.2011.10.785.  Google Scholar  S. Kesavan, Nonlinear Functional Analysis. A First Course, Texts and Readings in Mathematics 28, Hindustan Book Agency, 2004. Google Scholar  A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal., 12 (1988), 761-775.  doi: 10.1016/0362-546X(88)90037-5.  Google Scholar  N. S. Papageorgiou and F. Papalini, Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries, Israel Journal of Mathematics, 201 (2014), 761-796.  doi: 10.1007/s11856-014-1050-y.  Google Scholar  P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, no. 65. AMS, Providence, R. I. (1986). doi: 10.1090/cbms/065.  Google Scholar  P. H. Rabinowitz, J. Su and Z-Q Wang, Multiple solutions of superlinear elliptic equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), 97-108. Google Scholar
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