Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity

Abstract

We consider the semilinear elliptic equationwhere Ω is an open subset of R N (N 2) containing the origin and h is locally Lipschitz continuous on [0, ∞), positive in (0, ∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q ∈ (1, C N ) (that is, lim u→∞ h(λu)/ h(u) = λ q , for every λ > 0), where C N denotes either N/(N − 2) if N 3 or ∞ if N = 2. Our result extends a well-known theorem of Véron for the case h(u) = u q .

Isolated singularities for weighted quasilinear elliptic equations

“…Moreover, even in the special case p = 2 and b ≡ 1, this new method is much simpler to use than the earlier perturbation method of [4]. In Section 2 by assuming two facts (to be validated later in Section 3 and Section 7), we prove Theorem 1.4.…”

“…Motivated by [7,3,17] and our recent work [4], we classify here all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form They obtained a complete classification of the behavior near zero for all positive solutions when p − 1 < q < (p−1)N N −p (any q > p − 1 if p = N ). The homogeneity of the power non-linearity and various scaling arguments were key ingredients in the approach of [7] and other related papers such as [18,19,3].…”

“…We recently made progress in [4] by treating such a non-linearity h in the special case b ≡ 1 and p = 2. More exactly, we extended Véron's classification result in [18,19] to positive solutions of u = h(u) in Ω * when h ∈ RV q with q > 1.…”

“…These were used to obtain the precise limiting behavior of the solutions u with a strong singularity at zero. But the perturbation method in [4] seems difficult to apply if p = 2.…”

On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator

“…The reduction procedure relies on Step 3 and the construction of w ∞ in Step 2. We devise the super-solutions (sub-solutions) in Step 1 (Step 5) inspired by the work in [6] for μ = 0.…”