On the moving plane method for boundary blow-up solutions to semilinear elliptic equations

: We consider weak solutions to − ∆ u = f ( u ) on Ω 1 \ Ω 0 , with u = c ≥ 0 in ∂ Ω 1 and u = +∞ on ∂ Ω 0 , and we prove monotonicity properties of the solutions via the moving plane method. We also prove the radial symmetry of the solutions in the case of annular domains.


Introduction
Let Ω 0 and Ω 1 be two bounded smooth domains of ℝ N , N ≥ 2, such that Ω 0 ⊂ Ω 1 . Moreover, let f ∈ C 1 ([0, +∞)) and c ≥ 0. We consider weak solutions to the problem i.e., we consider u ∈ C 1 (Ω 1 \ Ω 0 ) such that and lim When c = 0, actually we deal with the case of positive solutions. Necessary and sufficient conditions for the existence of solutions to (1.1) are provided by the classical results of Keller [18] and Osserman [19], under suitable assumptions on the nonlinearity. The literature regarding boundary blow-up solutions is really wide (see, for example, [1-7, 12-17, 20, 21]). Here we exploit an adaptation of the celebrated moving plane technique (see [8]) in order to obtain monotonicity properties of the solutions to (1.1). The domain that we consider is not convex and the solutions are not in H 1 0 (Ω 1 \ Ω 0 ) as in the classical case. This is the same difficulty that occurs when dealing with the study of the uniqueness, symmetry and monotonicity properties of solutions to singular semilinear elliptic equations, see [9][10][11]. These problems exhibit in fact some similarities with problem (1.1) although the proofs cannot be adapted to our case.
In the second part of the paper, we prove the radial symmetry of the solutions on annular domains, under suitable assumptions. In our setting, this cannot be done just using the moving plane method, since the domain is not convex, and we prove that the solution is radially symmetric showing directly that the angular derivative is zero. The technique is based on a refined maximum principle for the linearized equation.
Let us introduce some notations. Let ν be a direction in ℝ N , with |ν| = 1. Given a real number λ, we set , that is, the reflection of x trough the hyperplane T ν λ . We will make the following assumption throughout the paper: (A) Ω 0 and Ω 1 are strictly convex with respect to the ν-direction and symmetric with respect to T ν 0 . Moreover, we set Observe that, by assumption (A), it follows that Ω ν λ is nonempty and We are now ready to state our main results.
Consequently, it follows that u is strictly increasing with respect to the ν-direction in the set In order to get symmetry results for the solution to (1.1), we restrict our attention to annular domains. We denote by B R the open ball of center 0 and radius R > 0 in ℝ N . By Theorem 1.1, we immediately deduce the following.
In the following we state sufficient conditions in order to deduce the radial symmetry of the solution, once we prove the monotonicity. We set v = −x ⋅ ∇u, and we denote by u θ the angular derivative of u.

Proof of Theorem 1.3
We start by proving the following proposition.
In the following we will also exploit the fact that as it follows by direct computation.
Set now t 0 = inf{t > 0 : u θ + tv ≥ 0 in B R 1 \ B R 0 }. We need to prove that t 0 = 0. Conversely, suppose t 0 > 0. By the definition of t 0 and Proposition 3.1, we obtain By the strong maximum principle, since u θ = 0 and v > 0 on ∂B R 1 , we get Since (1.3) is in force, there exists δ 0 > 0 such that Moreover, we have By continuity, for ε > 0 small, we have that Resuming, we have that