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 $0\in \Omega$. right here, $b(x)$ is a favorable non-stop functionality on $\overline \Omega\setminus\{0\}$ which behaves close to the foundation as an often various functionality at 0 with index $\theta$ more than $-2$. The nonlinearity $h$ is thought 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 confident suggestions of equation (0.1) whilst $h$ 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 $\xi>0$). 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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Extra resources for A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations With Inverse Square Potentials
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1) with γ = 0 has always a unique solution, which 1,α (B ∗ ) for some α ∈ (0, 1). 2 in [17]. 37 38 5. 1. 1. 4. 2. 2) 0 < lim inf |x|→0 > 0. If u u(x) u(x) ≤ lim sup − < ∞. Φ− (x) Φ |x|→0 λ λ (x) Proof. 5) with lim|x|→0 u(x)/Φ+ λ (x) = 0. Without any loss of generality, we can assume that B1 (0) ⊂ Ω. 2) into four steps. 9, as shown in Step 1. 2) is more difficult, being achieved in Steps 2–4. 10, there exists a positive constant C such that b(x) h(u) ≤ C|x|θ Lb (|x|) (h2 (u) + u) for 0 < |x| ≤ 1. 4) −Δv − 2 v + C|x|θ Lb (|x|) (h2 (v) + v) = 0 for 0 < |x| < 1.
This solution has the property that u(r)/Ψ− (r) is increasing and its behaviour is given in (b) above. Proof. 19) y (s) = φ(s)[y(s)]q for s ∈ (0, ∞) with φ(s) := b0 (e−s )e−s[N +2−q(N −2)]/2 . (a) Since lims→∞ y (s) exists, we have lims→∞ y(s)/s ∈ (0, ∞) if and only if lims→∞ y (s) ∈ (0, ∞). 19) has positive solutions satisfying lims→∞ y(s)/s ∈ (0, ∞) if and only if ∞ sq φ(s) ds < ∞, 0 which is equivalent to limτ →0 F1 (τ, ∞ ) < ∞. 13) holds, that is sq+1 φ(s) ds < ∞, 0 then y(s) = cs + d + o(1) as s → ∞ for some constants c and d with c > 0.
Indeed, the C 1 (0, 1]–function r (m+3)(p−N +2)+2N −2 b0 (r) is regularly varying at 0 with index (q + 3)(p − N + 2) + 2N − 2 + θ which is negative (since q ≥ q ∗ and p < (N − 2)/2). 57) r 1−(N −2−p)(m−1) b0 (r) dr = ∞. 56), we would + get limr→0 U∞ (r)/Φ+ λ (r) = 0. 52). We end the proof by constructing U∞ . 5). 58) ⎪ ⎩ U (x) = max u(y) for |x| = 1/n and |x| = 1. |y|=|x| Clearly, Un must be radially symmetric. 58). 9, we get u ≤ Un ≤ Un+1 in An for every n ≥ 2. 56) satisfying u ≤ U∞ in B ∗ . 8.
A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations With Inverse Square Potentials by Florica C. Cirstea
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