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‘Large’ solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour. (English) Zbl 0802.35038

From the introduction: We consider the equation ${\Delta }u=f\left(u\right)$ in ${\Omega }$, where ${\Omega }$ is a domain in ${ℝ}^{N}$ whose boundary is a compact ${C}^{2}$ manifold and $f$ is a positive differentiable function in ${ℝ}_{+}$ such that $f\left(0\right)=0$ and ${f}^{\text{'}}\ge 0$ everywhere. A solution $u$ satisfying $u\left(x\right)\to \infty$ as $x\to \partial {\Omega }$ is called a large solution. We are interested in the questions of existence and uniqueness of large solutions and in their asymptotic behaviour near the boundary.

More generally, we consider equations of the form ${\Delta }u=g\left(x,u\right)$ in ${\Omega }$, which includes the case $g\left(x,u\right)=h\left(x\right){u}^{p}$ where $p>1$ and $h$ is a positive continuous function in $\overline{{\Omega }}$ such that $h$ and $1/h$ are bounded. For this class of equations we describe the precise asymptotic behaviour of large solutions near the boundary and establish the uniqueness of such solutions.

##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35B40 Asymptotic behavior of solutions of PDE
##### References:
 [1] [BL] C. Bandle and H. Leutwiler,On a quasilinear elliptic equation and a Riemannian metric invariant under Möbius transformations, to appear. [2] [BM1] C. Bandle and M. Marcus,Un théorème de comparaison pour un problème elliptique avec une non-linéarité singulière, C. R. Acad. Sci. Paris287 Série A (1978), 861–863. [3] [BM2] C. Bandle and M. Marcus,Sur les solutions maximales de problèmes elliptiques nonlinéaires: bornes isopérimétriques et comportement asymptoique, C. R. Acad. Sci. Paris311 Série I (1990), 91–93. [4] [K] J. B. Keller,On solutions of ${\Delta }$u=f(u), Comm. Pure Appl. Math. 10 (1957), 503–510. · Zbl 0090.31801 · doi:10.1002/cpa.3160100402 [5] [Ko] N. Korevaar,Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J.32 (1983), 603–614. · Zbl 0528.35011 · doi:10.1512/iumj.1983.32.32042 [6] [LN] C. Loewner and L. Nirenberg,Partial differential equations invariant under conformal or projective transformations, inContributions to Analysis, ed. L. Ahlfors, Academic Press, New York, 1974, pp. 245–272. [7] [O] R. Osserman,On the inequality ${\Delta }$u(u), Pacific J. Math.7 (1957), 1641–1647. [8] [PS] L. E. Payne and I. Stackgold,Nonlinear problems in nuclear reactor analysis, Springer Lecture Notes in Math. #322 (1972), 298–301. [9] [S] R. Sperb,Maximum Principles and their Applications, Academic Press, New York, 1981.