Fundamental properties and asymptotic shapes of the singular and classical radial solutions for supercritical semilinear
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Nonlinear Differential Equations and Applications NoDEA
Fundamental properties and asymptotic shapes of the singular and classical radial solutions for supercritical semilinear elliptic equations Yasuhito Miyamoto
and Y¯ uki Naito
Abstract. We study singular radial solutions of the semilinear elliptic equation Δu + f (u) = 0 on finite balls in RN with N ≥ 3. We assume that f satisfies either f (u) = up + o(up ) with p > (N + 2)/(N − 2) or f (u) = eu + o(eu ) as u → ∞. We provide the existence and uniqueness of the singular radial solution, and show the convergence of regular radial solutions to the singular solution. Some applications to the bifurcation diagram of an elliptic Dirichlet problem are also given. Our results generalize and improve some known results in the literature. Mathematics Subject Classification. 35J61, 35A05, 35A24. Keywords. Semilinear elliptic equation, Singular solution, Supercritical, Uniqueness, Existence, Asymptotic shape.
1. Introduction We study singular radial solutions of the semilinear elliptic equation Δu + f (u) = 0
in Ω \ {0},
N
where Ω ⊂ R , with N ≥ 3, is a finite ball centered with the origin and f ∈ C 1 [0, ∞). In the study, we consider solutions to the ordinary differential equation N −1 u + f (u) = 0 for r > 0. (1.1) u + r By a singular solution u∗ (r) of (1.1) we mean that u∗ (r) is a classical solution of the Eq. (1.1) for 0 < r ≤ r0 with some r0 > 0 and it satisfies u∗ (r) → ∞ as r → 0. For α > 0, we denote by u(r, α) a regular solution of (1.1) satisfying u(0) = α and u (0) = 0. In (1.1) we assume that f (u) satisfies either f (u) = up + o(up ) or
f (u) = eu + o(eu ) as u → ∞,
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Y. Miyamoto and Y. Naito
NoDEA
where p > pS = (N + 2)/(N − 2). It is well known that the singular solution plays a key role in the study of the solution structure of the supercritical problem (1.1). See, e.g. [13,17,26,27,30]. In this paper, we prove the existence and uniqueness of the singular solution of (1.1), and also the convergence of the regular solution u(r, α) to the singular solution as α → ∞. These properties have been studied extensively in the literature, because of various potential applications for both elliptic and parabolic problems. When f (u) = up , (1.1) has the exact singular solution u∗ (r) = Ar−2/(p−1) if p > N/(N − 2), where 1/(p−1) 2 2 . (1.2) A= N −2− p−1 p−1 It was shown by Serrin–Zou [34, Proposition 3.1] that, if p > pS , the singular solution of (1.1) is unique. On the other hand, it was shown in [7,34] that, if N/(N −2) < p < pS , (1.1) has a continuum of singular solutions. When f (u) = up +u with p > pS , it was shown by Merle–Peletier [24] that (1.1) has a singular solution and the regular solution u(r, α) converges to the singular solution as α → ∞. See also [4]. When f (u) = up − u, Chern et al. [8] investigated the entire structure of radial solutions according to their behaviors at the origin and infinity, and showed that (1.1) possesses a unique singular solution in the case N > 10 and p is large. In [26] t
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