# A Complete Classification of the Isolated Singularities for by Florica C. Cirstea

By Florica C. Cirstea

During this paper, the writer considers semilinear elliptic equations of the shape $-\Delta u- \frac{\lambda}{|x|^2}u +b(x)\,h(u)=0$ in $\Omega\setminus\{0\}$, the place $\lambda$ is a parameter with $-\infty<\lambda\leq (N-2)^2/4$ and $\Omega$ is an open subset in $\mathbb{R}^N$ with $N\geq three$ such that $0\in \Omega$. right here, $b(x)$ is a favorable non-stop functionality on $\overline \Omega\setminus\{0\}$ which behaves close to the starting place as an often various functionality at 0 with index $\theta$ more than $-2$. The nonlinearity $h$ is thought non-stop on $\mathbb{R}$ and confident on $(0,\infty)$ with $h(0)=0$ such that $h(t)/t$ is bounded for small $t>0$. the writer thoroughly classifies the behaviour close to 0 of all optimistic options of equation (0.1) whilst $h$ is frequently various at $\infty$ with index $q$ more than $1$ (that is, $\lim_{t\to \infty} h(\xi t)/h(t)=\xi^q$ for each $\xi>0$). particularly, the author's effects follow to equation (0.1) with $h(t)=t^q (\log t)^{\alpha_1}$ as $t\to \infty$ and $b(x)=|x|^\theta (-\log |x|)^{\alpha_2}$ as $|x|\to 0$, the place $\alpha_1$ and $\alpha_2$ are any actual numbers

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2), we ﬁnd that 1 f (K(|x0 |)) for every x0 satisfying 0 < |x0 | ≤ r0 . C On the other hand, since limr→0 K(r) = ∞ and f ∈ RVq−1 (∞), we infer that there exists a constant C1 > 0 such that f (u(x0 )) ≤ f (K(r)) ≤ Cf (C1 K(r)) for every r ∈ (0, r0 ]. Consequently, we have f (u(x0 )) ≤ f (C1 K(|x0 |)) for every x0 satisfying 0 < |x0 | ≤ r0 . 1). Proof of Claim. A simple calculation shows that for x ∈ B|x0 |/2 (x0 ), we have ΔP(x) = 16 C J (|x0 |) f 2 (P) |x − x0 |2 N ζ(x) + 16 |x0 |2 f (P) |x0 |2 3 f (P)f (P) − 2 [f (P)]2 .

34) ΔV(r) (ξ) + 2(2 − N + p)∇V(r) (ξ) · 2 = r 2+p |ξ|N −2−p b(rξ) h(u(rξ)) |ξ| for 0 < |ξ| < r0 /r. For every ﬁxed ξ ∈ RN \ {0}, we have 0 < |ξ| < r0 /r provided that r > 0 is suﬃciently small. 35) lim r 2+p b(rξ) h(u(rξ)) = 0 for every ξ ∈ RN \ {0}. 23), it suﬃces to see that H(r) := r 2+θ+p Lb (r) h2 (C1 Φ− λ (r)) → 0 as r → 0. This follows immediately if λ ≤ 0, since p ≤ 0 and h2 (t)/t is bounded for t > 0 small so that H is bounded from above by a regularly varying function at zero with positive index (θ + 2).

8) holds. Using that lim sup|x|→0 u(x) = ∞, there exists a sequence (rn )n≥1 which decreases to zero such that max u(x) ≥ CC2 |x|=rn for every n ≥ 1. Without loss of generality, we assume that 0 < rn < r0 for all n ≥ 1. 8), we have u(x) ≥ C for |x| = rn and all n ≥ 1. 5) in Br0 (0) \ {0} with h2 instead of h. 9 on each annulus {x ∈ RN : rn < |x| < r1 } with n ≥ 2, we conclude that u(x) ≥ C for all 0 < |x| ≤ r1 . 18). Hence, we ﬁnd lim|x|→0 u(x) = ∞. 3. 9). 12. 34 4. 9 (Regularity). Fix r0 > 0 small such that B4r0 (0) ⊂ Ω.