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
References:
[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.  Google Scholar

[2]

H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc., 85 (1982), 591-595.  doi: 10.2307/2044072.  Google Scholar

[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.  Google Scholar

[4]

A. Castro, Métodos de reducción via minimax, Primer Simposio Colombiano de Análisis Funcional, Medellín, Colombia, (1981). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Springer Verlag, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509.   Google Scholar

[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.  Google Scholar

[16]

S. Kesavan, Nonlinear Functional Analysis. A First Course, Texts and Readings in Mathematics 28, Hindustan Book Agency, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

References:
[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.  Google Scholar

[2]

H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc., 85 (1982), 591-595.  doi: 10.2307/2044072.  Google Scholar

[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.  Google Scholar

[4]

A. Castro, Métodos de reducción via minimax, Primer Simposio Colombiano de Análisis Funcional, Medellín, Colombia, (1981). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Springer Verlag, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math., 45 (1986), 501-509.   Google Scholar

[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.  Google Scholar

[16]

S. Kesavan, Nonlinear Functional Analysis. A First Course, Texts and Readings in Mathematics 28, Hindustan Book Agency, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[2]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

[3]

Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure & Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507

[4]

Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003

[5]

M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013

[6]

Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617

[7]

Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083

[8]

Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, William D. Kalies, Dinesh Kasti. Efficient computation of Lyapunov functions for Morse decompositions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2419-2451. doi: 10.3934/dcdsb.2015.20.2419

[9]

Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319

[10]

Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010

[11]

Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985

[12]

Joseph A. Iaia. Localized radial solutions to a semilinear elliptic equation in $\mathbb{R}^n$. Conference Publications, 1998, 1998 (Special) : 314-326. doi: 10.3934/proc.1998.1998.314

[13]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795

[14]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011

[15]

Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399

[16]

Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122

[17]

Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107

[18]

Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092

[19]

Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747

[20]

Bo Li, Hongwei Lou. Cesari-type conditions for semilinear elliptic equation with leading term containing controls. Mathematical Control & Related Fields, 2011, 1 (1) : 41-59. doi: 10.3934/mcrf.2011.1.41

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (8)
  • Cited by (0)

Other articles
by authors

[Back to Top]