Abstract
In this work we investigate the existence and asymptotic behavior of positive solution to the quasilinear elliptic equation
with homogeneous Neumann condition. Here we show the existence of least-energy solution u λ for large λ and that the maximum of u λ concenters around a point of ∂Ω.
Research partially supported by PADCT, CAPES and the Millennium Institute for the Global Advancement of Brazilian Mathematics-IM-AGIMB.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity. Nonlinear analysis, 9–25, Quaderni, Scuola Norm. Sup., Pisa, 1991.
Adimurthi, Filomena 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), no. 2
T. Aubin, Nonlinear Analysis on Monge-Ampère Equations, Springer-Verlag, New York, (1982).
P. Cherrier, Meilleures constantes dans des inégalités relatives aux espaces de Sobolev. (French) [Best constants in inequalities involving Sobolev spaces]. Bull. Sci. Math. (2) 108 (1984) 225–262.
E.A.M. de Abreu, J.M.B do Ó and E.S. Medeiros, Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems. Nonlinear Analysis 60 (2005) 1443–1471.
M. del Pino and C. Flores, Asymptotics of Sobolev embeddings and singular perturbations for the p-Laplacian. Proc. Amer. Math. Soc. 130 (2002) 2931–2939.
E.F. Keller and L.A. Segal, Initiation of Slime mold aggregation viewed as an instability, J. Theory. Biol., 26 (1970) 399–415.
M.G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12 (1988) 1203–1219.
C.S. Lin, W.M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72 (1988) 1–27.
P.L. Lions, The Concentration Compactness principle in the Calculus of Variations, The limit case (Part 1 and Part 2), Rev. Mat. Iberoamericana 1 (1985) 145–201, 45–121.
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.
P. Pucci and J. Serrin, A general variational identity. Indiana Univ. Math. J. 35 (1986) 681–703.
P. Tolksdorf, Regularity for a More General Class of Quasilinear Elliptic Equations, J. Diff. Equations 51 (1984) 126–150.
J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984) 191–202.
X.J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations 93 (1991) 283–310.
M. Willem, Minimax theorems, Birkhäuser Boston, Inc., Boston, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Djairo Guedes de Figueiredo on his 70th birthday
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Medeiros, E.S. (2005). On the Shape of Least-Energy Solutions to a Quasilinear Elliptic Equation Involving Critical Sobolev Exponents. In: Cazenave, T., et al. Contributions to Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 66. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7401-2_26
Download citation
DOI: https://doi.org/10.1007/3-7643-7401-2_26
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7149-4
Online ISBN: 978-3-7643-7401-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)