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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 \(\Delta u = f(u)\) in \(\Omega\), where \(\Omega\) is a domain in \(\mathbb{R}^ N\) whose boundary is a compact \(C^ 2\) manifold and \(f\) is a positive differentiable function in \(\mathbb{R}_ +\) such that \(f(0) = 0\) and \(f' \geq 0\) 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.
More generally, we consider equations of the form \(\Delta u = g(x,u)\) in \(\Omega\), which includes the case \(g(x,u) = h(x)u^ p\) 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. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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