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

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 good non-stop functionality on $\overline \Omega\setminus\{0\}$ which behaves close to the starting place as a frequently various functionality at 0 with index $\theta$ more than $-2$. The nonlinearity $h$ is believed non-stop on $\mathbb{R}$ and optimistic 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 strategies of equation (0.1) while $h$ is often various at $\infty$ with index $q$ more than $1$ (that is, $\lim_{t\to \infty} h(\xi t)/h(t)=\xi^q$ for each $\xi>0$). specifically, the author's effects observe 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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Extra resources for A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations With Inverse Square Potentials

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8) max u(x) ≤ C2 min u(x) |x|=r for every r ∈ (0, r0 ]. |x|=r Proof. 5 in [49]. We give it here for the sake of completeness. ∗ := B2r0 (0) \ {0}. Let y ∈ RN be such that 0 < |y| ≤ r0 . 9) Lu = Δu + d(x)u in Ωy := B2|y|/3 (y), where d(x) is defined by d(x) := λ b(x) h(u(x)) ∗ for x ∈ B2r − . 6) in [25] are satisfied with λ = 1, Λ = N and ν(y) = supx∈Ωy |d(x)|. 11) with C0 = C0 (N ). We next show that η(y) ν(y) is bounded above by a constant independent of y and u. 12) b2 (x) := b(x) h(u(x)) u(x) for every x ∈ RN with 0 < |x| ≤ 2r0 .

A) Eq. 1) admits positive solutions satisfying limr→0 u(r)/Ψ+ (r) ∈ (0, ∞) if and only if limτ →0 F1 (τ, ) < ∞. 14) u(r) = c Ψ+ (r) + d Ψ− (r) + o(Ψ− (r)) as r → 0. (b) For any number γ > 0, Eq. 1) with u(1) = γ admits a unique positive solution with limr→0 u(r)/Ψ+ (r) = 0. 15) ) < ∞ and in this case u(r) = α0 + [(α0 )q + o(1)] Ψ− (r) r b0 (ζ)ζ N −q(N −2) 2 log(r/ζ) dζ 0 as r → 0+ . 1) with limr→0 ui (r)/Ψ+ (r) = 0 will satisfy limr→0 u1 (r)/u2 (r) = 1 and limr→0 ui (r)/Ψ− (r) = 0. 2. THE CRITICAL PARAMETER 27 (c) Assume that limτ →0 F1 (τ, ) < ∞.

2. 3. 5) satisfies lim|x|→0 u(x) = ∞. 7. 2. 18) for some λ = λ1 with 0 < λ1 < (N − 2)2 /4. 4. 5) such that γ − := lim sup |x|→0 ) < ∞ for some small > 0. 22) u(x) = γ− − |x|→0 Φ (x) λ lim and x · ∇u(x) = −p γ − . 5) in Ω. Proof. 1]. Both [17] and [22] treat quasilinear elliptic equations without a singular potential. We provide all the details since our proof here brings in new distinctions due to the inverse square potential. We fix r0 > 0 small such that B2r0 (0) ⊂ Ω. 23) u(x) ≤ C1 |x|−p for every 0 < |x| ≤ 2r0 .

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