October  2016, 36(10): 5595-5626. doi: 10.3934/dcds.2016046

On the Hollman McKenna conjecture: Interior concentration near curves

1. 

Instituto de Matemáticas, UNAM, Circuito Exterior S/N, Coyoácan, Cd, Universitaria, 04510 Ciudad de México, D.F., Mexico

2. 

Department of Basic Mathematics, Centro de Investigacióne en Mathematicás, Guanajuato, Mexico

Received  September 2015 Revised  December 2015 Published  July 2016

Consider the problem \begin{equation} \notag \left\{\begin{aligned} -\epsilon^2\Delta u&=|u|^p-\Phi_{1} &&\text{in } \Omega\\ u &= 0 &&\text{on }\partial \Omega \end{aligned} \right. \end{equation} where $\epsilon>0$ is a parameter, $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ and $p>2$. Let $\Gamma$ be a stationary non-degenerate closed curve relative to the weighted arc-length $\int_{\Gamma} \Phi_{1}^{\frac{p+3}{2p}}.$ We prove that for $\epsilon>0$ sufficiently small, there exists a solution $u_{\epsilon}$ of the problem, which concentrates near the curve $\Gamma$ whenever $d(\Gamma, \partial \Omega)>c_{0}>0.$ As a result, we prove the higher dimensional concentration for a Ambrosetti-Prodi problem, thereby proving an affirmative result to the conjecture by Hollman-McKenna [9] in two dimensions.
Citation: Bhakti Bhusan Manna, Sanjiban Santra. On the Hollman McKenna conjecture: Interior concentration near curves. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5595-5626. doi: 10.3934/dcds.2016046
References:
[1]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces,, Ann. Mat. Pura Appl., 93 (1972), 231. doi: 10.1007/BF02412022. Google Scholar

[2]

B. Breuer, P. J. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof,, J. Differential Equations, 195 (2003), 243. doi: 10.1016/S0022-0396(03)00186-4. Google Scholar

[3]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture,, J. Differential Equations, 210 (2005), 317. doi: 10.1016/j.jde.2004.07.017. Google Scholar

[4]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture: Part II,, Comm. in Partial Differential Equations, 30 (2005), 1331. doi: 10.1080/03605300500258865. Google Scholar

[5]

E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case,, Adv. Differential Equations, 12 (2007), 961. Google Scholar

[6]

M. del Pino and C. Munőz, The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity,, J. Differential Equations, 231 (2006), 108. doi: 10.1016/j.jde.2006.07.003. Google Scholar

[7]

M. Del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135. Google Scholar

[8]

de Djairo G. Figueiredo, S. Santra and P. Srikanth, Non-radially symmetric solutions for a superlinear Ambrosetti-Prodi type problem in a ball,, Commun. Contemp. Math., 7 (2005), 849. doi: 10.1142/S0219199705001982. Google Scholar

[9]

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. doi: 10.3934/cpaa.2011.10.785. Google Scholar

[10]

A. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem,, J. Math. Anal. Appl., 84 (1981), 282. doi: 10.1016/0022-247X(81)90166-9. Google Scholar

[11]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth,, Calc. Var PDE., 28 (2007), 471. doi: 10.1007/s00526-006-0051-z. Google Scholar

[12]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth: Part 2,, J. Differential Equations, 227 (2006), 301. doi: 10.1016/j.jde.2006.02.011. Google Scholar

[13]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators,, Mathematics and its Applications (Soviet Series), (1991). doi: 10.1007/978-94-011-3748-5. Google Scholar

[14]

A. Malchiodi and M. Montenegro, Boundary Concentration Phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar

[15]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5. Google Scholar

[16]

F. Mahmoudi, A. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve,, Comm. Pure Appl. Math., 62 (2009), 1155. doi: 10.1002/cpa.20290. Google Scholar

[17]

R. Molle and D. Passaseo, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities,, J. Funct. Anal., 259 (2010), 2253. doi: 10.1016/j.jfa.2010.05.010. Google Scholar

[18]

J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case,, J. Funct. Anal., 244 (2007), 639. doi: 10.1016/j.jfa.2006.11.002. Google Scholar

[19]

J. Wei and J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains,, Indiana Univ. Math. J., 56 (2007), 3025. doi: 10.1512/iumj.2007.56.3133. Google Scholar

show all references

References:
[1]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces,, Ann. Mat. Pura Appl., 93 (1972), 231. doi: 10.1007/BF02412022. Google Scholar

[2]

B. Breuer, P. J. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof,, J. Differential Equations, 195 (2003), 243. doi: 10.1016/S0022-0396(03)00186-4. Google Scholar

[3]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture,, J. Differential Equations, 210 (2005), 317. doi: 10.1016/j.jde.2004.07.017. Google Scholar

[4]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture: Part II,, Comm. in Partial Differential Equations, 30 (2005), 1331. doi: 10.1080/03605300500258865. Google Scholar

[5]

E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case,, Adv. Differential Equations, 12 (2007), 961. Google Scholar

[6]

M. del Pino and C. Munőz, The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity,, J. Differential Equations, 231 (2006), 108. doi: 10.1016/j.jde.2006.07.003. Google Scholar

[7]

M. Del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135. Google Scholar

[8]

de Djairo G. Figueiredo, S. Santra and P. Srikanth, Non-radially symmetric solutions for a superlinear Ambrosetti-Prodi type problem in a ball,, Commun. Contemp. Math., 7 (2005), 849. doi: 10.1142/S0219199705001982. Google Scholar

[9]

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. doi: 10.3934/cpaa.2011.10.785. Google Scholar

[10]

A. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem,, J. Math. Anal. Appl., 84 (1981), 282. doi: 10.1016/0022-247X(81)90166-9. Google Scholar

[11]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth,, Calc. Var PDE., 28 (2007), 471. doi: 10.1007/s00526-006-0051-z. Google Scholar

[12]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth: Part 2,, J. Differential Equations, 227 (2006), 301. doi: 10.1016/j.jde.2006.02.011. Google Scholar

[13]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators,, Mathematics and its Applications (Soviet Series), (1991). doi: 10.1007/978-94-011-3748-5. Google Scholar

[14]

A. Malchiodi and M. Montenegro, Boundary Concentration Phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar

[15]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5. Google Scholar

[16]

F. Mahmoudi, A. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve,, Comm. Pure Appl. Math., 62 (2009), 1155. doi: 10.1002/cpa.20290. Google Scholar

[17]

R. Molle and D. Passaseo, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities,, J. Funct. Anal., 259 (2010), 2253. doi: 10.1016/j.jfa.2010.05.010. Google Scholar

[18]

J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case,, J. Funct. Anal., 244 (2007), 639. doi: 10.1016/j.jfa.2006.11.002. Google Scholar

[19]

J. Wei and J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains,, Indiana Univ. Math. J., 56 (2007), 3025. doi: 10.1512/iumj.2007.56.3133. Google Scholar

[1]

Elisa Sovrano. Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019

[2]

F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355

[3]

Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184

[4]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761

[5]

Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465

[6]

Mingqi Xiang, Binlin Zhang. A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 413-433. doi: 10.3934/dcdss.2019027

[7]

Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378

[8]

Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099

[9]

Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099

[10]

David Colton, Lassi Päivärinta, John Sylvester. The interior transmission problem. Inverse Problems & Imaging, 2007, 1 (1) : 13-28. doi: 10.3934/ipi.2007.1.13

[11]

Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure & Applied Analysis, 2016, 15 (2) : 563-576. doi: 10.3934/cpaa.2016.15.563

[12]

Shuangjie Peng, Jing Zhou. Concentration of solutions for a Paneitz type problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1055-1072. doi: 10.3934/dcds.2010.26.1055

[13]

Naoufel Ben Abdallah, Irene M. Gamba, Giuseppe Toscani. On the minimization problem of sub-linear convex functionals. Kinetic & Related Models, 2011, 4 (4) : 857-871. doi: 10.3934/krm.2011.4.857

[14]

Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111

[15]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079

[16]

Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417

[17]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

[18]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[19]

Kyril Tintarev. Positive solutions of elliptic equations with a critical oscillatory nonlinearity. Conference Publications, 2007, 2007 (Special) : 974-981. doi: 10.3934/proc.2007.2007.974

[20]

Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]