Read e-book online A Complete Classification of the Isolated Singularities for PDF

By Florica C. Cirstea

ISBN-10: 0821890220

ISBN-13: 9780821890226

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 $0\in \Omega$. the following, $b(x)$ is a favorable 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 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 recommendations of equation (0.1) while $h$ 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 genuine numbers

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Example text

9), we infer that v∞ (r) ≤ cΦ− λ (r) for every r ∈ (Rn , R1 ) and every integer n ≥ 2. Since limn→∞ Rn = 0, we find that v∞ (r) ≤ cΦ− λ (r) for every r ∈ (0, R1 ). 7). This completes Step 2. Step 3. 8) lim sup u(x) = ∞ when 0 < λ < (N − 2)2 /4. 8) does not use the assumption that limτ →0 I ∗∗ (τ, ) < ∞. 5) when 0 < λ < (N − 2)2 /4, which will be relevant for later. 9) 2 ⎪ ⎩ lim sup u(x) < ∞ when 0 < λ < (N − 2) /4. 10, we find that h2 (v∞ (x)) ≤ Cv∞ (x) for all 0 < |x| ≤ 1, where C > 0 is a large constant.

1) with limr→0 ui (r)/Φ+ λ (r) = 0 will satisfy limr→0 ui (r)/Φ− (r) = 0 and lim u (r)/u (r) = 1. r→0 1 2 λ (c) Assume that limτ →0 I1 (τ, ) < ∞. 1) with u(1) = γ and limr→0 u(r)/Φ+ λ (r) ∈ (0, ∞). 4) + ⎪ ⎩ lim u(r)/Φλ (r) = ∞ and u(1) = γ. 4) has a unique positive solution for every γ > 0. 5) r (q+3)(2p−N +2) b1 (r) 1. If limτ →0 I1 (τ, positive solution 2. If limτ →0 I1 (τ, positive solution whose behaviour ≤0 for 0 < r ≤ 1. 4). 1) with u(1) = γ, namely the positive solution is given in (b) above.

5. 4(d), the condition limτ →0 F1 (τ, ) < ∞ is always satisfied. 20) ⎪ ⎩ lim U(r)/Ψ+ (r) = ∞. 19). 19)). 6. 1) with limr→0 u(r)/Ψ+ (r) = ∞ (respectively, limr→0 u(r)/Φ+ λ (r) = ∞). 1(e2)) is sufficient to ensure Case (B). 1(e2)) does not hold without imposing some additional requirement on b0 . 1 in [17]. 5) by modifying the ideas in [17] to take into account the inverse square potential. We rely on regular variation theory and use standard techniques going back to works of Friedman–V´eron [22] and V´eron [49].

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A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations With Inverse Square Potentials by Florica C. Cirstea


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