Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

We study the semilinear elliptic equation \begin{equation*} -\Delta u=u^\alpha |\log u|^\beta\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2}<\alpha<\frac{n+2}{n-2}$ and $-\infty<\beta<\infty$. Our main result establishes that nonnegative solution $u\in C^2(B_1\setminus\{0\})$ of the above equation either has a removable singularity at the origin or behaves like \begin{equation*} u(x) = A(1+o(1)) |x|^{-\frac{2}{\alpha-1}} \left(\log \frac{1}{|x|}\right)^{-\frac{\beta}{\alpha-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2}{\alpha-1}\right)^{1-\beta}\left(n-2-\frac{2}{\alpha-1}\right)\right]^{\frac{1}{\alpha-1}}. \end{equation*}


Introduction
Let n ≥ 3 and B 1 be the unit open ball in ℝ n . This paper is concerned with the behavior of nonnegative solutions of −∆u = u α |log u| β in B 1 \ {0}, (1.1) where α and β are real numbers satisfying n n − 2 < α < n + 2 n − 2 and −∞ < β < ∞. (1.2) We say that u is a nonnegative solution of (1.1) if u ∈ C 2 (B 1 \ {0}) is nonnegative and satisfies (1.1) pointwise. In addition, we say that a nonnegative solution u of (1.1) is singular if u is unbounded in any punctured ball B r \ {0}, with 0 < r < 1.
The case β = 0 in (1.1) is by now well understood; in their pioneering work [4], Gidas and Spruck established a series of results that completely characterize the asymptotic behavior of local solutions of (1.1) (with β = 0). The main goal of this paper is to obtain similar results for (1.1) when the exponents α and β are in the range given by (1.2).
Our main result is the following. Theorem 1.1. Assume α and β satisfy (1.2) and let u be a nonnegative solution of (1.1). Then the following alternative holds: (i) either u has a removable singularity at the origin, (ii) or u is a singular solution and satisfies For β = 0, we recover the result in [4,Theorem 1.3]. Let us note that in the case β = 0, the approach in [4] relies to a large extend on the properties of the scaling function u λ (x) = λ The asymptotic behavior of nonnegative singular solutions has been studied in various settings. In addition to the classical results [4] and [1], Korevaar et al. [6] derived the improved asymptotic behavior of the nonnegative singular solutions of −∆u = u n+2 n−2 by a more geometric approach. Meanwhile, C. Li [7] extended the result on the asymptotic radial symmetry of singular solutions of −∆u = g(u) for a more general g(u) considered in [1]. Recently, the asymptotic radial symmetry has been achieved for other operators, such as conformally invariant fully nonlinear equations [5,8], fractional equations [2], and fractional p-laplacian equations [3].
This paper extends the classical argument in [4] and [1] to a log-type nonlinearity. One of the key observations is that from the asymptotic radial symmetry achieved in [1] for nonnegative solutions of −∆u = g(u), one can obtain an optimal asymptotic upper bound for g (u) u . Hence, we are left with preserving the optimality by transforming g (u) u to u under a suitable inverse mapping. This observation indeed allows us to consider a more general class of equations of the type where f is a slowly varying function at infinity, under some additional assumptions. A typical example is where k i are positive integers, β i are real numbers and log (k) u = log(log (k−1) u) for k ≥ 2 with log (1) u = log u. However, we shall not specify the additional assumptions for the nonlinearity f as they turn out to involve technical and cumbersome computations. Hence, we present the argument only with f(u) = |log u| β in order to simplify the presentation.
Throughout the paper, we shall write where C > 0 depends at most on n, α and β. We shall also use the notation

Asymptotic behavior around a non-removable singularity
Letū (r) denote the spherical average of u on the ball of radius r, that is, The following result is a slight modification of [1, Theorem 1.1].

Theorem 2.1. Let u be a nonnegative solution of
with an isolated singularity at the origin. Suppose that g(t) is a locally Lipschitz function, which in a neighborhood of infinity satisfies the conditions below: The original result in [1, Theorem 1.1] requires condition (i) above to be satisfied for all t > 0, but a careful analysis of its proof shows that this condition is enough to hold in a neighborhood of infinity. It is not hard to see that The next lemma provides an asymptotic upper bound forū . Lemma 2.2. We haveū as r → 0.
Hence, from the assumptionū (r) → ∞ as r → 0 and the factū (r) < 0, it follows that Note that for any sufficiently large s satisfying 2|β| ≤ (α − 1) log s, we have whence we may proceed from the integral above as 1 u α−1 (r)(logū (r)) β ≥ cr 2 for sufficiently small r > 0. Thus, we arrive at Setting w(s) to be the inverse function¹ of se s , we know that s w(s) is the inverse function of s log s. Since t α−1 (log t) β = (cs log s) β , with s = t α−1 β , we deduce from (2.5) and the choice of w that However, since log s − log log s ≤ w(s) ≤ log s for sufficiently large s, we arrive at (2.2).
Let us next define with t = − log r and θ ∈ n−1 .
Let us defineψ Averaging (2.7) over n−1 , we obtain for large t. Lemma 2.4. We have as t → ∞.
Proof. In this proof, C > 0 will depend on n only and may differ from one line to another.  Therefore, it follows from the interior gradient estimates that Using this observation along with (2.1), (2.5) and the above gradient estimate, we find |∇(u −ū )| ≤ C(ū + r 2ūα |logū | β ) ≤ Cū on ∂B r . (2.21) Since   On the other hand, sinceψψ = 1 2 (ψ 2 ) −ψ 2 , a further integration by parts produces To this end, we shall pass to the limit in (2.17) with t → ∞. Note that from (2.9) we have Although we do not know yet ifψ(t) converges as t → ∞, we still know from (2.20) that it converges along a subsequence. Denoting byψ 0 a limit value ofψ(t) along a subsequence, say t = t j → ∞, after passing to the limit in (2.17), with t = t j , we obtain, from (2.22), (2.27) and (2.33), that Thus, in view of (1.4), we haveψ Now the continuity ofψ implies thatψ(t) converges as t → ∞ (without extracting any subsequence) either to 0 or A. If there are two distinct sequences t j → ∞ and t j → ∞ such thatψ(t j ) → 0 andψ(t j ) → A, then by the intermediate value theorem, there must exist some other t j → ∞ such thatψ(t j ) → A 2 , which violates (2.34). Thus, the proof is completed.
We are now in a position to prove Theorem 1.1. Proof of Theorem 1.1. If (2.23) is true, then, in view of (2.15), we observe that Hence, from (2.1), we derive (1.3) and (1.4), which establishes the proof for Theorem 1.1 (ii).
The rest of the argument follows closely that of the proof in [1,Theorem 1.3].
Hence, we are only left with the case β < 0. Since u(x) = (1 + O(r))ū (r), (2.35) implies that for each q ≥ 1, for some constant C > 0 depending on n, α, β and q. Therefore, u α |log u| β ∈ L p (B 1 ) for any p ≥ 1, and in particular for p > n. This implies that ∆u ∈ L p (B 1 ) for p > n, so u ∈ C 1,α (B 1/2 ) for α = 1 − n p , proving again that the origin is a removable singularity. Thus, the proof of Theorem 1.1 (i) is completed.