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Zbl 0802.35038
Bandle, Catherine; Marcus, Moshe
`Large' solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour.
(English)
[J] J. Anal. Math. 58, 9-24 (1992). ISSN 0021-7670; ISSN 1565-8538/e

From the introduction: We consider the equation $\Delta u = f(u)$ in $\Omega$, where $\Omega$ is a domain in $\bbfR\sp N$ whose boundary is a compact $C\sp 2$ manifold and $f$ is a positive differentiable function in $\bbfR\sb +$ such that $f(0) = 0$ and $f' \ge 0$ everywhere. A solution $u$ satisfying $u(x) \to \infty$ 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.\par More generally, we consider equations of the form $\Delta u = g(x,u)$ in $\Omega$, which includes the case $g(x,u) = h(x)u\sp p$ 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 2000:
*35J60 Nonlinear elliptic equations
35B05 General behavior of solutions of PDE
35B40 Asymptotic behavior of solutions of PDE

Keywords: semilinear elliptic equations; asymptotic behavior near the boundary

Cited in: Zbl 0817.35027

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