General uniqueness results for large solutions
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
General uniqueness results for large solutions Juli´ an L´ opez-G´omez, Luis Maire and Laurent V´eron Abstract. We give a series of very general sufficient conditions in order to ensure the uniqueness of positive large solutions for −Δu + f (x, u) = 0 in a bounded domain Ω where f : Ω × R → R+ is a continuous function, such that f (x, 0) = 0 for x ∈ Ω, and f (x, r) > 0 for x in a neighborhood of ∂Ω and all r > 0. Mathematics Subject Classification. 35 J 61, 31 B 15, 28 C 05. Keywords. Keller–Osserman condition, local graph condition, large solutions, maximum principle, strong barrier property.
Contents 1. 2. 3. 4.
Introduction Proof of Theorem 1.1 Proof of Theorem 1.2 Appendix 4.1. On the Keller–Osserman condition 4.2. On the strong barrier property References
1. Introduction Let Ω ⊂ RN be a bounded domain and f : Ω × R → R+ a continuous function such that f (x, 0) = 0 and r → f (x, r) is nondecreasing for x ∈ Ω, and f (x, r) > 0 for x in a neighborhood of ∂Ω and all r > 0. This paper deals with the uniqueness question of the solution of the equation −Δu + f (x, u) = 0
in Ω,
(1.1)
satisfying the blow-up condition lim u(x) = ∞,
d(x)→0
(1.2)
where d(x) := dist (x, ∂Ω). Whenever a solution to (1.1)–(1.2) exists, it is called a large solution or an explosive solution. Although, thanks to [22, Corollary 3.3], in the one-dimensional case N = 1 with f (x, u) ≡ f (u) the above problem admits a unique solution, the question of ascertaining whether or not (1.1)–(1.2) possesses a unique solution received only partial answers even in the autonomous case when f (x, u) ≡ f (u) is independent of x ∈ Ω. Astonishingly, when N = 1 the large solution can be unique even when f (u) is somewhere decreasing (see [21,22]), which measures the real level of difficulty of the problem of characterizing the set of f (x, u) for which (1.1)–(1.2) has a unique positive solution; it is an extremely challenging problem. 0123456789().: V,-vol
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Page 2 of 14
J. L´ opez-G´ omez, L. Maire and L. V´ eron
ZAMP
Existence of large solutions is associated to the Keller–Osserman condition. When f is independent of x, this condition was introduced in [12,30] for proving the first existence results of large solutions in a smooth bounded domain. It reads ∞ s ds < ∞ for some a > 0 where F (s) = f (t)dt. (1.3) F (s) − F (a) 0
a
It is pointed out in [8] that the monotonicity condition is not needed for the existence of a large solution in a ball. Actually the result therein is more precise since it makes the distinction between the existence of a large solution in some ball and the existence in every ball by introducing the Sharpened Keller–Osserman condition, which asserts that the integral in (1.3) tends to 0 when a tends to ∞. Existence and uniqueness of large solutions of more general equations such as aij (x)uxi xj + bi (x)uxi + c(x)u + f (u) = h(x), − (1.4) i,j
i
even with singular lower-order coefficients and unbounded forcing term h has been investigated in [27,29] and [34].
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