Abstract
In this paper we study the number of the boundary single peak solutions of the problem
for \({\varepsilon}\) small and p subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.
Similar content being viewed by others
References
Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on R n. Progress in Mathematics, vol. 240. Birkhäuser, Boston (2006)
Azorero J.G., Malchiodi A., Montoro L., Peral I.: Concentration of solutions for some singularly perturbed mixed problems: existence results. Arch. Ration. Mech. Anal. 196, 907–950 (2010)
Berestycki H., Lions P.L.: Nonlinear scalar field equations I and II. Arch. Ration. Mech. Anal. 82, 313–375 (1983)
DelPino M., Felmer P., Wei J.: On the role of mean curvature in some singularly perturbed Neumann problems. SIAM J. Math. Anal. 31, 63–79 (1999)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)
Grossi M.: On the number of single-peak solutions of the nonlinear Schrödinger equation. Ann. I. H. Poincaré AN 19(3), 261–280 (2002)
Gui C.: Multipeak solutions for a semilinear Neumann problem. Duke Math. J. 84, 739–769 (1996)
Kwong M.K.: Uniqueness of positive solutions of Δu − u + u p = 0 in \({\mathbb {R}^n}\) . Arch. Ration. Mech. Anal. 105, 243–266 (1989)
Li Y.Y.: On a singularly perturbed equation with Neumann boundary condition. Commun. Partial Differ. Equ. 23(3), 487–545 (1998)
Li Y., Zhao C.: Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem. J. Math. Anal. Appl. 336, 1368–1383 (2007)
Lin C.-S., Ni W.-M., Takagi I.: Large amplitude stationary solutions to a chemotaxis system. J. Differ. Equ. 72, 1–27 (1988)
Malchiodi, A.: Concentration of Solutions for Some Singularly Perturbed Neumann Problems. Geometric Analysis and PDEs. Lecture Notes in Mathematics, vol. 1977, CIME (2009)
Ni W.-M., Takagi I.: On the shape of least-energy solutions to a semilinear Neumann problem. Commun. Pure Appl. Math. 44, 819–851 (1991)
Ni W.-M., Takagi I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247–281 (1993)
Wei J.: On the boundary spike layer solutions to a singularly perturbed Neumann problem. J. Differ. Equ. 134, 104–133 (1997)
Wei J.: Uniqueness and critical spectrum of boundary spike solutions. Proc. R. Soc. Edinb. Sect. A 131, 1457–1480 (2001)
Wei J., Winter M.: Higher-order energy expansions and spike locations. Calc. Var. 20, 403–430 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Malchiodi.
Rights and permissions
About this article
Cite this article
Grossi, M., Neves, S.L.N. Exact multiplicity results for a singularly perturbed Neumann problem. Calc. Var. 48, 713–737 (2013). https://doi.org/10.1007/s00526-012-0569-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-012-0569-1