Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T23:24:12.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of multiple positive solutions for a semilinear equation with critical exponent*

Published online by Cambridge University Press:  14 November 2011

Deng Yinbing
Deng Yinbing
Affiliation:
Department of Mathematics, Huazhong Normal University, Wuhan 430070, P.R. China
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we discuss the problem

We show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 1-2 , 1992 , pp. 161 - 175
Copyright
Copyright © Royal Society of Edinburgh 1992

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

1Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
2Ambrosetti, A. and Struwe, M.. A note on the problem −Δu = λu + u|u|2*−2. Manuscripta Math. 54 (1968), 373–379.CrossRefGoogle Scholar
3Bahri, A. and Berestycki, H.. A perturbation method in critical point theory and applications. Trans. Amer. Math. Soc. 267 (1981), 1–32.CrossRefGoogle Scholar
4Benci, V. and Cerami, G.. Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rational Mech. Anal. 99 (1987), 283–300.CrossRefGoogle Scholar
5Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations, (I) Existence of a ground state; (II) Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82 (1983), 313–375.CrossRefGoogle Scholar
6Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
7Cerami, G., Solomin, S. and Struwe, M.. Some existence results for superlinear elliptic boundary value problems involving Sobolev exponents. J. Fund. Anal. 69 (1986), 289–306.CrossRefGoogle Scholar
8de Figueiredo, D. G.. On superlinear Ambrosetti-Prodi problem. Nonlinear Anal. 8 (1984), 655–665.CrossRefGoogle Scholar
9Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry of positive solutions of nonlinear elliptic equations in RN. Adv. in Math. Stud. 7A (1981), 369–402.Google Scholar
10Lions, P. L.. On positive solutions of semilinear elliptic equations in unbounded domains. In Nonlinear Diffusion Equations and Their Equilibrium States, eds Ni, W. M., Peletier, L. A. and Serrin, J. (New York/Berlin: Springer, 1989).Google Scholar
11Stuart, C. A.. Bifurcation in LP(RN) for a semilinear elliptic equation. Proc. London Math. Soc. 57(1988), 511–541.CrossRefGoogle Scholar
12Zhu, X. P. and Yang, J. F.. Quasilinear elliptic equations on unbounded domain involving critical Sobolev exponents. J. Partial Differential Equations 2(2) (1989), 53–64.Google Scholar
13Zhu, X. P. and Zhou, H. S.. Existence of multiple positive solutions of an inhomogeneous semilinear elliptic problem in unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 115A (1990), 301–318.CrossRefGoogle Scholar