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Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations. (English) Zbl 0877.35042
The authors study properties of positive solutions of $$\Delta u+ hu-ku^p= f\tag1$$ in a (possibly) nonsmooth $N$-dimensional domain $\Omega$, $N\ge 2$, subject to the condition $$u(x)\to\infty\quad\text{if}\quad \delta(x):= \text{dist}(x,\partial\Omega)\to 0.\tag2$$ Here $p>1$ and $h$, $k$, $f$ are continuous in $\overline\Omega$ with $k>0$ and $f\ge 0$. Positive solutions of (1) satisfying (2) are called large solutions. A central point of this paper is the following localization principle: let $\Omega$ be a (not necessarily bounded) domain having the graph property and suppose $u$ is a positive solution of (1) satisfying $u(x)\to\infty$ locally uniformly as $x\to\Gamma$, where $\Gamma\subset\partial\Omega$ is relatively open. If $v$ is a large solution, then $v(x)/u(x)\to 1$ locally uniformly as $x\to\Gamma$. Closely related to this is a uniqueness result for large solutions in bounded domains having the graph property. For bounded Lipschitz domains the authors prove the existence of positive constants $c_1\le c_2$ such that the (unique) large solution $u$ of (1) satisfies $c_1\delta(x)^{-{2\over p-1}}\le u(x)\le c_2\delta(x)^{-{2\over p-1}}$ for all $x\in\Omega$. This is also a consequence of the localization principle and an existence theorem, obtained for large solutions in bounded domains satisfying the exterior cone condition. If the domain is not Lipschitz, the rate of blow-up at the boundary may be lower. This is proved for domains having a re-entrant cusp in the case $p\ge(N-1)/(N-3)$. Finally, the authors discuss the dependence of large solutions on the function $k$ and the domain $\Omega$.

35J60Nonlinear elliptic equations
35J67Boundary values of solutions of elliptic equations
35B40Asymptotic behavior of solutions of PDE
Full Text: DOI Numdam EuDML
[1] Benguria, R.; Brezis, H.; Lieb, E.: The Thomas-Fermi-von weizsaäcker theory of atoms and molecules. Comm. math. Phys. 79, 167-180 (1981) · Zbl 0478.49035
[2] Bandle, C.; Marcus, M.: Sur LES solutions maximales de problèmes elliptiques non linéaires. C. R. Acad. sci. Paris 311, 91-93 (1990) · Zbl 0726.35041
[3] Bandle, C.; Marcus, M.: Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior. Jl. d’analyse math. 58, 9-24 (1992) · Zbl 0802.35038
[4] Bandle, C.; Marcus, M.: Large solutions of semilinear elliptic equations with singular coefficients. Pitman R. N. series 244, 25-38 (1992) · Zbl 0795.35023
[5] C. Bandle and M. Marcus, Asymptotic behavior of solutions and their derivatives for semilinear elliptic problems with blowup on the boundary, Ann. Inst. Poincaré (to appear). · Zbl 0840.35033
[6] Brezis; Veron, L.: Remouvable singularities for some nonlinear elliptic equations. Arch. rat. Mech. anal. 75, 1-6 (1980)
[7] Gidas, B.; Ni, W.; Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. math. Phys. 68, 209-243 (1979) · Zbl 0425.35020
[8] Keller, J. B.: On solutions of ${\Delta}$u = f (u). Comm. pure appl. Math. 10, 503-510 (1957) · Zbl 0090.31801
[9] Lee, J. N.; Parker, T. H.: The yamabe problem. Bull. amer. Math. soc. 17, 37-91 (1987) · Zbl 0633.53062
[10] J. F. Le Gall, A path-valued Markov process and its connections with partial differential equations, Proc. 1st European Congress of Mathematics, Birkhäuser (to appear).
[11] Marcus, M.: On solutions with blow up at the boundary for a class of semilinear elliptic equations. Developments in PDE and applications to mathematical physiscs, 65-77 (1993)
[12] Marcus, M.; Veron, L.: Uniqueness of solutions with blow up at the boundary for a class of nonlinear elliptic equations. C. R. Acad. sci. Paris 317, 559-563 (1993) · Zbl 0803.35041
[13] Osserman, R.: On the inequality ${\Delta}$u = f (u). Pacific jl. Math. 7, 1641-1647 (1957) · Zbl 0083.09402
[14] Pinchover, Y.: Criticality and ground states for second order elliptic equations. Jl. diff. Equ. 80, 237-250 (1989) · Zbl 0697.35036
[15] Pinsky, R.: Positive harmonic functions and diffusion. Cambridge studies in advanced math. 45 (1995)
[16] Veron, L.: Semilinear elliptic equations with uniform blow up on the boundary. Jl. d’analyse math. 59, 231-250 (1992)
[17] Veron, L.: Comportement asymptotique des solutions d’équations elliptiques semi linéaires dans RN. Ann. mat. Pura appl. 127, 25-50 (1981)
[18] Veron, L.: Singular solutions of some nonlinear elliptic equations. Nonlinear anal. T. M. & A. 5, 225-242 (1981)