# American Institute of Mathematical Sciences

June  2016, 36(6): 3035-3076. doi: 10.3934/dcds.2016.36.3035

## Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China 2 Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago 3 Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago

Received  March 2015 Revised  October 2015 Published  December 2015

In this paper we consider the following problem \begin{eqnarray} \label{abstract} \quad \left\{ \begin{array}{ll}-\Delta u +u= u^{{n-k+2\over n-k-2} \pm\epsilon} & \mbox{ in } \Omega \\ u>0& \mbox{ in }\Omega                                  (0.1)\\ {\partial u\over\partial\nu}=0 & \mbox{ on } \partial\Omega \end{array} \right. \end{eqnarray} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\ge 7$, $k$ is an integer with $k\ge 1$, and $\epsilon >0$ is a small parameter. Assume there exists a $k$-dimensional closed, embedded, non degenerate minimal submanifold $K$ in $\partial \Omega$. Under a sign condition on a certain weighted avarage of sectional curvatures of $\partial \Omega$ along $K$, we prove the existence of a sequence $\epsilon = \epsilon_j \to 0$ and of solutions $u_\epsilon$ to (0.1) such that $$|\nabla u_\epsilon |^2 \, \rightharpoonup \, S \delta_K , \quad {\mbox {as}} \quad \epsilon \to 0$$ in the sense of measure, where $\delta_K$ denotes a Dirac delta along $K$ and $S$ is a universal positive constant.
Citation: Shengbing Deng, Fethi Mahmoudi, Monica Musso. Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3035-3076. doi: 10.3934/dcds.2016.36.3035
##### References:
 [1] Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi. Scuola Norm. Sup. Pisa, (1991), 9-25. [2] Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations, 20 (1995), 591-631. doi: 10.1080/03605309508821110. [3] Adimurthi, F. Pacella and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., 113 (1993), 318-350. doi: 10.1006/jfan.1993.1053. [4] Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and L1- estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Diff. Integ. Equ., 8 (1995), 41-68. [5] W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, Journal of Differential Equations, 251 (2011), 881-901. doi: 10.1016/j.jde.2011.05.009. [6] W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM J. Math. Anal., 43 (2011), 2519-2541. doi: 10.1137/100812100. [7] G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q. [8] J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity, J. Differential Equations, 244 (2008), 2473-2497. doi: 10.1016/j.jde.2008.02.024. [9] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [10] D. Cao and T. Kupper, On the existence of multipeaked solutions to a semilinear Neumann problem, Duke Math. J., 97 (1999), 261-300. doi: 10.1215/S0012-7094-99-09712-0. [11] E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241. [12] M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898. doi: 10.1512/iumj.1999.48.1596. [13] M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (2000), 63-79. doi: 10.1137/S0036141098332834. [14] M. del Pino, F. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents, J. Eur. Math. Soc. (JEMS), 16 (2014), 1687-1748. doi: 10.4171/JEMS/473. [15] M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553-1605. doi: 10.4171/JEMS/241. [16] M. del Pino, M. Musso and A. Pistoia, Supercritical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincare Anal. Non-Linearie, 22 (2005), 45-82. doi: 10.1016/j.anihpc.2004.05.001. [17] N. Ghoussoub and C. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z., 229 (1998), 443-474. doi: 10.1007/PL00004663. [18] N. Ghoussoub, C. Gui and M. Zhu, On a singularly perturbed Neumann problem with the critical exponent, Comm. Partial Differential Equations, 26 (2001), 1929-1946. doi: 10.1081/PDE-100107812. [19] M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations, 11 (2000), 143-175. doi: 10.1007/PL00009907. [20] C. Gui, Multi-peak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769. doi: 10.1215/S0012-7094-96-08423-9. [21] C. Gui and C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, J. Reine Angew. Math., 546 (2002), 201-235. doi: 10.1515/crll.2002.044. [22] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27. doi: 10.1016/S0022-0396(99)80016-3. [23] Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations, 23 (1998), 487-545. doi: 10.1080/03605309808821354. [24] C.-S. Lin, Locating the peaks of solutions via the maximum principle, I. The Neumann problem, Comm. Pure Appl. Math., 54 (2001), 1065-1095. doi: 10.1002/cpa.1017. [25] F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281. doi: 10.1002/cpa.20139. [26] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Diff. Equat., 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [27] F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525. doi: 10.1016/j.aim.2006.05.014. [28] 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. [29] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16 (2006), 924-958. doi: 10.1007/s00039-006-0566-7. [30] F. Mahmoudi, F. S. Sanchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture in higher dimension, J. Differential Equations, 258 (2015), 243-280. doi: 10.1016/j.jde.2014.09.010. [31] S. Maier-Paape, K. Schmitt and Z. Q. Wang, On Neumann problems for semilinear elliptic equations with critical nonlinearity existence and symmetry of multi-peaked solutions, Comm. Partial Differential Equations, 22 (1997), 1493-1527. doi: 10.1080/03605309708821309. [32] 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. [33] 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. [34] A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom. Funct. Anal., 15 (2005), 1162-1222. doi: 10.1007/s00039-005-0542-7. [35] R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom., 13 (2005), 633-670. doi: 10.4310/CAG.2005.v13.n4.a1. [36] M. Musso and J. Yang, Curve like concentration layers for a singularly perturbed nonlinear problem with critical exponents, Comm. Partial Differential Equations, 39 (2014), 1048-1103. doi: 10.1080/03605302.2013.851215. [37] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [38] W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary Partial Differential Equations. Handbook Differential Equations, I, North-Holland, Amsterdam, 2004, 157-233. doi: 10.1016/S1874-5733(04)80005-6. [39] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. doi: 10.1002/cpa.3160440705. [40] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4. [41] W.-M. Ni, X.-B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents, Duke Math. J., 67 (1992), 1-20. doi: 10.1215/S0012-7094-92-06701-9. [42] S. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f (u) =0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. [43] O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains, Comm. Contemp. Math., 1 (1999), 405-449. doi: 10.1142/S0219199799000158. [44] O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc. (JEMS), 7 (2005), 449-476. doi: 10.4171/JEMS/35. [45] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (IV), 110 (1976), 353-372. doi: 10.1007/BF02418013. [46] L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615. doi: 10.1090/S0002-9947-10-04955-X. [47] Z. Q. Wang, High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent, Proc. Roy. Soc. Edimburgh, 125 (1995), 1003-1029. doi: 10.1017/S0308210500022617. [48] X.-J. Wang, Neumann problem of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z. [49] J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133. doi: 10.1006/jdeq.1996.3218.

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##### References:
 [1] Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi. Scuola Norm. Sup. Pisa, (1991), 9-25. [2] Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations, 20 (1995), 591-631. doi: 10.1080/03605309508821110. [3] Adimurthi, F. Pacella and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., 113 (1993), 318-350. doi: 10.1006/jfan.1993.1053. [4] Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and L1- estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Diff. Integ. Equ., 8 (1995), 41-68. [5] W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, Journal of Differential Equations, 251 (2011), 881-901. doi: 10.1016/j.jde.2011.05.009. [6] W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM J. Math. Anal., 43 (2011), 2519-2541. doi: 10.1137/100812100. [7] G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q. [8] J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity, J. Differential Equations, 244 (2008), 2473-2497. doi: 10.1016/j.jde.2008.02.024. [9] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [10] D. Cao and T. Kupper, On the existence of multipeaked solutions to a semilinear Neumann problem, Duke Math. J., 97 (1999), 261-300. doi: 10.1215/S0012-7094-99-09712-0. [11] E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241. [12] M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898. doi: 10.1512/iumj.1999.48.1596. [13] M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (2000), 63-79. doi: 10.1137/S0036141098332834. [14] M. del Pino, F. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents, J. Eur. Math. Soc. (JEMS), 16 (2014), 1687-1748. doi: 10.4171/JEMS/473. [15] M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553-1605. doi: 10.4171/JEMS/241. [16] M. del Pino, M. Musso and A. Pistoia, Supercritical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincare Anal. Non-Linearie, 22 (2005), 45-82. doi: 10.1016/j.anihpc.2004.05.001. [17] N. Ghoussoub and C. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z., 229 (1998), 443-474. doi: 10.1007/PL00004663. [18] N. Ghoussoub, C. Gui and M. Zhu, On a singularly perturbed Neumann problem with the critical exponent, Comm. Partial Differential Equations, 26 (2001), 1929-1946. doi: 10.1081/PDE-100107812. [19] M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations, 11 (2000), 143-175. doi: 10.1007/PL00009907. [20] C. Gui, Multi-peak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769. doi: 10.1215/S0012-7094-96-08423-9. [21] C. Gui and C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, J. Reine Angew. Math., 546 (2002), 201-235. doi: 10.1515/crll.2002.044. [22] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27. doi: 10.1016/S0022-0396(99)80016-3. [23] Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations, 23 (1998), 487-545. doi: 10.1080/03605309808821354. [24] C.-S. Lin, Locating the peaks of solutions via the maximum principle, I. The Neumann problem, Comm. Pure Appl. Math., 54 (2001), 1065-1095. doi: 10.1002/cpa.1017. [25] F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281. doi: 10.1002/cpa.20139. [26] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Diff. Equat., 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [27] F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525. doi: 10.1016/j.aim.2006.05.014. [28] 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. [29] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16 (2006), 924-958. doi: 10.1007/s00039-006-0566-7. [30] F. Mahmoudi, F. S. Sanchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture in higher dimension, J. Differential Equations, 258 (2015), 243-280. doi: 10.1016/j.jde.2014.09.010. [31] S. Maier-Paape, K. Schmitt and Z. Q. Wang, On Neumann problems for semilinear elliptic equations with critical nonlinearity existence and symmetry of multi-peaked solutions, Comm. Partial Differential Equations, 22 (1997), 1493-1527. doi: 10.1080/03605309708821309. [32] 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. [33] 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. [34] A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom. Funct. Anal., 15 (2005), 1162-1222. doi: 10.1007/s00039-005-0542-7. [35] R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom., 13 (2005), 633-670. doi: 10.4310/CAG.2005.v13.n4.a1. [36] M. Musso and J. Yang, Curve like concentration layers for a singularly perturbed nonlinear problem with critical exponents, Comm. Partial Differential Equations, 39 (2014), 1048-1103. doi: 10.1080/03605302.2013.851215. [37] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [38] W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary Partial Differential Equations. Handbook Differential Equations, I, North-Holland, Amsterdam, 2004, 157-233. doi: 10.1016/S1874-5733(04)80005-6. [39] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. doi: 10.1002/cpa.3160440705. [40] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4. [41] W.-M. Ni, X.-B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents, Duke Math. J., 67 (1992), 1-20. doi: 10.1215/S0012-7094-92-06701-9. [42] S. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f (u) =0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. [43] O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains, Comm. Contemp. Math., 1 (1999), 405-449. doi: 10.1142/S0219199799000158. [44] O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc. (JEMS), 7 (2005), 449-476. doi: 10.4171/JEMS/35. [45] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (IV), 110 (1976), 353-372. doi: 10.1007/BF02418013. [46] L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615. doi: 10.1090/S0002-9947-10-04955-X. [47] Z. Q. Wang, High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent, Proc. Roy. Soc. Edimburgh, 125 (1995), 1003-1029. doi: 10.1017/S0308210500022617. [48] X.-J. Wang, Neumann problem of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z. [49] J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133. doi: 10.1006/jdeq.1996.3218.
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