zbMATH — the first resource for mathematics

Elliptic equations with nearly critical growth. (English) Zbl 0657.35058
The author studies what happens to the solution of the boundary value problem \[ (1)\quad -\Delta u=u^ p\quad u>0\quad in\quad \Omega \in R^ N;\quad u=0\quad on\quad \partial \Omega \] if p approaches the critical Sobolev exponent \(p=(N+2)/(N-2)\) (if \(p\geq (N+2)/(N-2)\), the problem (1) has no solutions) and \(\Omega =B_ R=\{x\in R^ N:\) \(x<R\}\). He uses ODE arguments to obtain precise asymptotic estimates as \(p\to (N+2)/(N-2)=0\).
Reviewer: J.H.Tian

35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Abramowitz, M; Stegun, I.A, ()
[2] Atkinson, F.V; Peletier, L.A, Emden-Fowler equations involving critical exponents, Nonlinear anal. TMA, 10, 755-776, (1986) · Zbl 0662.34024
[3] Atkinson, F.V; Peletier, L.A, Elliptic equations with critical growth when N ⩾ 3 and N = 2, () · Zbl 0669.35038
[4] \scF. V. Atkinson and L. A. Peletier, “Large Solutions of Semilinear Elliptic Equations with Critical Exponents,” in press. · Zbl 0682.35042
[5] Brezis, H, Some variational problems with lack of compactness, (), 165-201
[6] Brezis, H; Nirenberg, L, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. pure appl. math., 36, 437-477, (1983) · Zbl 0541.35029
[7] \scC. Budd, Semilinear Elliptic Equations with Near Critical Growth Rates, Proc. Roy Soc. Edinburgh, to appear. · Zbl 0662.35003
[8] Fowler, R.H, The form near infinity of real continuous solutions of a certain differential equation on the second order, Quart. J. math. Cambridge ser., 45, 289-350, (1914) · JFM 45.0479.01
[9] Fowler, R.H, Further studies of Emden’s and similar differential equations, Quart. J. math. Oxford ser., 2, 259-288, (1931) · Zbl 0003.23502
[10] Gidas, B; Ni, Wei-Ming; Nirenberg, L, Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[11] Lions, P.L, On the existence of positive solutions of semilinear elliptic equations, SIAM rev., 24, 441-467, (1982) · Zbl 0511.35033
[12] \scJ. B. McLeod and J. Norbury, in preparation.
[13] \scS. I. Pohozaev, Eigenfunctions of the equation Δu + λ|(u) = 0, Dokl. Akad. Nauk SSSR\bf165, 36-39 (in Russian) · Zbl 0141.30202
[14] Rabinowitz, P.H, Variational methods for nonlinear eigenvalue problems, Indiana univ. math. J., 23, 729-754, (1974) · Zbl 0278.35040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.