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 and Continuous Dynamical Systems, 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., (4) 93 (1972), 231-246. doi: 10.1007/BF02412022.

[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), 243-269. doi: 10.1016/S0022-0396(03)00186-4.

[3]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351. doi: 10.1016/j.jde.2004.07.017.

[4]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture: Part II, Comm. in Partial Differential Equations, 30 (2005), 1331-1358. doi: 10.1080/03605300500258865.

[5]

E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993.

[6]

M. del Pino and C. Munőz, The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity, J. Differential Equations, 231 (2006), 108-134. doi: 10.1016/j.jde.2006.07.003.

[7]

M. Del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.

[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), 849-866. doi: 10.1142/S0219199705001982.

[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), 785-802. doi: 10.3934/cpaa.2011.10.785.

[10]

A. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294. doi: 10.1016/0022-247X(81)90166-9.

[11]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var PDE., 28 (2007), 471-508. doi: 10.1007/s00526-006-0051-z.

[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), 301-332. doi: 10.1016/j.jde.2006.02.011.

[13]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), 59. Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3748-5.

[14]

A. Malchiodi and M. Montenegro, Boundary Concentration Phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1568. doi: 10.1002/cpa.10049.

[15]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143. doi: 10.1215/S0012-7094-04-12414-5.

[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), 1155-1264. doi: 10.1002/cpa.20290.

[17]

R. Molle and D. Passaseo, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities, J. Funct. Anal., 259 (2010), 2253-2295. doi: 10.1016/j.jfa.2010.05.010.

[18]

J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667. doi: 10.1016/j.jfa.2006.11.002.

[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), 3025-3073. doi: 10.1512/iumj.2007.56.3133.

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., (4) 93 (1972), 231-246. doi: 10.1007/BF02412022.

[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), 243-269. doi: 10.1016/S0022-0396(03)00186-4.

[3]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351. doi: 10.1016/j.jde.2004.07.017.

[4]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture: Part II, Comm. in Partial Differential Equations, 30 (2005), 1331-1358. doi: 10.1080/03605300500258865.

[5]

E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993.

[6]

M. del Pino and C. Munőz, The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity, J. Differential Equations, 231 (2006), 108-134. doi: 10.1016/j.jde.2006.07.003.

[7]

M. Del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146. doi: 10.1002/cpa.20135.

[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), 849-866. doi: 10.1142/S0219199705001982.

[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), 785-802. doi: 10.3934/cpaa.2011.10.785.

[10]

A. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294. doi: 10.1016/0022-247X(81)90166-9.

[11]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var PDE., 28 (2007), 471-508. doi: 10.1007/s00526-006-0051-z.

[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), 301-332. doi: 10.1016/j.jde.2006.02.011.

[13]

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), 59. Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3748-5.

[14]

A. Malchiodi and M. Montenegro, Boundary Concentration Phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1568. doi: 10.1002/cpa.10049.

[15]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143. doi: 10.1215/S0012-7094-04-12414-5.

[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), 1155-1264. doi: 10.1002/cpa.20290.

[17]

R. Molle and D. Passaseo, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities, J. Funct. Anal., 259 (2010), 2253-2295. doi: 10.1016/j.jfa.2010.05.010.

[18]

J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667. doi: 10.1016/j.jfa.2006.11.002.

[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), 3025-3073. doi: 10.1512/iumj.2007.56.3133.

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