# American Institute of Mathematical Sciences

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 \notag \left\{\begin{aligned} -\epsilon^2\Delta u&=|u|^p-\Phi_{1} &&\text{in } \Omega\\ u &= 0 &&\text{on }\partial \Omega \end{aligned} \right. 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 articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]