Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T01:22:20.406Z Has data issue: false hasContentIssue false
  • English
  • Français

A singularly perturbed elliptic problem involving supercritical Sobolev exponent*

Published online by Cambridge University Press:  14 November 2011

Yaotian Shen  and
Shusen Yan
Yaotian Shen
Affiliation:
Department of Applied Mathematics, South China University of Technology, Guangzhou, P.R. China
Shusen Yan
Affiliation:
Department of Applied Mathematics, South China University of Technology, Guangzhou, P.R. China
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with −Δu + εuq−1 = u2*−1, , where q > 2*, ε > 0. We first show that the minimiser of the associated minimisation problem blows up at the global minimum point of H(x, x), where H(y, x) is the regular part of the Green's function. We then prove that for each strictly local minimum point x0 of H(x, x), this problem has a solution concentrating at x0 as ε→0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 809 - 819
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adimurthi, F. Pacella and Yadava, S. L.. Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity. J. Fund. Anal. 113 (1993), 318–50.CrossRefGoogle Scholar
2Bahri, A. and Coron, J. M.. On a nonlinear elliptic equation involving the Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41 (1988), 253–94.CrossRefGoogle Scholar
3Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 38 (1983), 437–77.CrossRefGoogle Scholar
4Cao, D., Dancer, E. N., Noussair, E. and Yan, S.. On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete Contin. Dynam Systems 2 (1996), 221–36.CrossRefGoogle Scholar
5Cao, D., Noussair, E. and Yan, S.. Existence and uniqueness results on single-peaked solutions of a nonlinear problem. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear).Google Scholar
6Merle, F. and Peletier, L. A.. Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth, II the nonradial case. J. Fund. Anal. 105 (1992), 1–41.CrossRefGoogle Scholar
7Ni, W. M. and Takagi, I.. Locating the peaks of least energy solutions to a semilinear Neumann problem. Duke Math. J. 70 (1993), 247–81.CrossRefGoogle Scholar
8Ni, W. M. and Wei, J.. On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48 (1995), 731–68.CrossRefGoogle Scholar
9Pohozaev, S. I.. Eigenfunctions of the equation Δu + λf(u) = 0. Soviet Math. 6 (1969), 1408–11.Google Scholar
10Rey, O.. The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent. J. Fund. Anal. 89 (1990), 1–52.CrossRefGoogle Scholar
11Wei, J.. Locating the blow-up point of a semilinear Dirichlet problem involving critical Sobolev exponent. J. Math. Soc. Japan (to appear).Google Scholar
12Wei, J.. On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem. J. Differential Equations (to appear).Google Scholar