## 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

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
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### References:

 [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
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