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

  • * Corresponding author: G. Reyes

    * Corresponding author: G. Reyes

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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  • 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.

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


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  • [1] 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.
    [2] H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc., 85 (1982), 591-595.  doi: 10.2307/2044072.
    [3] B. BreuerP. 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.
    [4] A. Castro, Métodos de reducción via minimax, Primer Simposio Colombiano de Análisis Funcional, Medellín, Colombia, (1981).
    [5] A. Castro and J. Cossio, Multiple solutions for a nonlinear Dirichlet problem, SIAM J. Math. Anal., 25 (1994), 1554-1561.  doi: 10.1137/S0036141092230106.
    [6] A. CastroJ. 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.
    [7] 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.
    [8] 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.
    [9] 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.
    [10] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Springer Verlag, 1993. doi: 10.1007/978-1-4612-0385-8.
    [11] D. Clark, A variant of the Lyusternik-Schnirelmann theory, Indiana U. Math. J., 22 (1973), 65-74.  doi: 10.1512/iumj.1972.22.22008.
    [12] 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.
    [13] 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.
    [14] H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509. 
    [15] 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.
    [16] S. Kesavan, Nonlinear Functional Analysis. A First Course, Texts and Readings in Mathematics 28, Hindustan Book Agency, 2004.
    [17] 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.
    [18] 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.
    [19] 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.
    [20] P. H. RabinowitzJ. 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. 
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