Remarks on uniqueness of boundary blow-up solutions

Asymptotic Behavior and Uniqueness of Blow-up Solutions of Elliptic Equations

“…We start with a simple result, which was proved for a more general class of operators in [16,Theorem 1.2]. Here, we reproduce (for the reader's convenience) the proof of [12,Lemma 2.4], which is close in spirit (but not in detail) to the proof of [21, Lemma 3.1].…”

“…For example, [16] is concerned with the blow-up problem for the equation div(|Du| p−2 Du) = g(u), and [8,9,7] look at the equation ∆u + au = b(x)g(u) with a a suitable constant and b a nonnegative function satisfying some additional technical conditions relating b to g and a. We defer a study of such problems to a future work, but point out here that we are able to study g from a larger class of functions than in those works (when specialized to p = 2 in [16] and to b ≡ 1 and a = 0 in [8,9,7]). …”

“…When g(s)/s is not necessarily increasing, other methods have been used which rely on other conditions for g. (We refer to [28] and [29] for a discussion of these other methods.) In [16], the authors suggest that these structure conditions can be dispensed with, at least for smooth domains. Since there are known examples (see, e.g.…”

“…Since there are known examples (see, e.g. [10,20]) of nonsmooth domains in which uniqueness fails even for g(s) = s q with q > 1, we consider an intermediate smoothness situation, and we show how to implement the program of [16] in sufficiently smooth domains. In the applications, we shall see a relation between the smoothness of the domain and the structure of g in order to infer uniqueness, although we do not make any claims that our conditions are sharp.…”

“…Hence Theorem 6.1 again provides a uniqueness result in this case. (Note that, in [16], the corresponding structure is g(s) = e s + As q for some positive exponent q.) Now, let us suppose that there are positive constants t 1 and η ≤ θ such that h = ln(g) satisfies…”