Abstract
Let us consider the problem where Ω is a bounded domain in R N (N > 2) with a smooth boundary ∂Ω and g a function from R to R such that g(0) = 0. We consider the problem of existence of a solution of (I) when where λ,μ ∈ R and p > q > 1. In particular we consider the function although most of the results for (1.2) continue to hold when g merely satisfies (1.1)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ambrosetti, A. & P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381.
Atkinson, F.V. & L.A. Peletier, Emden-Fowler equations involving critical exponents, Nonlinear Anal. TMA. 10 (1986) 755–776.
Atkinson, F.V. & L.A. Peletier, Large solutions of elliptic equations involving critical exponents, Asymptotic Anal. 1 (1988) 139–160.
Atkinson, F.V. & L.A. Peletier, Elliptic equations with nearly critical growth, J. Diff. Equ. 70 (1987) 349–365.
Bahri, A. & J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988) 253–294.
Bandle, C. & L.A. Peletier, Nonlinear elliptic problems with critical exponent in shrinking annuli, Math. Ann. 280 (1988) 1–19.
Brezis, H. & L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477.
Brezis, H. & L.A. Peletier, Asymptotics for elliptic equations involving critical Sobolev exponents, In Partial differential equations and the calculus of variations, (Eds. F. Colombini, A. Marino, L. Modica & S. Spagnolo), pp. 149–192, Birkhäuser, 1989.
Budd, C, Applications of Shilnikov’s theory to semilinear elliptic equations, SIAM J. Math. Anal. 20 (1989) 1069–1080.
Budd, C. & J. Norbury, Semilinear elliptic equations and supercritical growth, J. Diff. Equ. 68 (1987) 169–197.
Glangetas, L., C.R.A.S.S. 312 (1991) 807–810.
Han, Zheng Chao, Thesis, Courant Institute, 1990.
Kwong, M.K., J. B. McLeod, L.A. Peletier & W.C. Troy, On ground state solutions of-Δu = u p-u q. To appear in J. Diff. Equ.
Lewandowski, R., Thesis, Paris, 1990.
Merle, F. & L.A. Peletier, Positive solutions of elliptic equations involving supercritical growth. Proc. Royal Soc. Edinburgh. 119A(1991) 49–62.
Merle, F. & L.A. Peletier, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth I. The radial case.. Arch. Rational Mech. Anal. 112 (1990) 1–19.
Merle, F. & L.A. Peletier, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth II. The general case.. To appear.
Pohozaev, S.I., Eigenfiuctions of the equation Δu + λf(u) = 0, Dokl. Akad. Nauk. SSSR 165 36–39 and Sov. Math. 6 (1965) 1408-1411.
Rabinowitz, P.H., Variational methods for nonlinear eigenvalue problems, Indiana Math. J. 23 (1974) 729–754.
Rey, O., The rôle of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89 (1990), 1–52.
Rey, O., Proof of two conjectures of H. Brezis and L.A. Peletier, Manuscripta Math. 65 (1989) 19–37.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media New York
About this chapter
Cite this chapter
Merle, F., Peletier, L.A. (1992). On Supercritical Phenomena. In: Lloyd, N.G., Ni, W.M., Peletier, L.A., Serrin, J. (eds) Nonlinear Diffusion Equations and Their Equilibrium States, 3. Progress in Nonlinear Differential Equations and Their Applications, vol 7. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0393-3_28
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0393-3_28
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6741-6
Online ISBN: 978-1-4612-0393-3
eBook Packages: Springer Book Archive