A Complete Classification of the Isolated Singularities for by Florica C. Cirstea 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 good 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 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 confident suggestions 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 $\xi>0$). 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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8. 60) lim inf − = 0. 52) holds with Φ− λ instead of Φλ . 61) θ+2 < m < q. p We have m > 1 since q ≥ q ∗∗ . 8 replacing Φ+ λ by Φ− . In particular, we define λ b0 (r) := c1 r θ Lb (r) χ(Φ− λ (r)) for 0 < r ≤ 1, which is a regularly varying function at zero with index θ − p(q − m) that is greater than −2. 56). 54) and limτ →0 I ∗∗ (τ, ) = ∞, we infer that r 1−p(m−1) b0 (r) dr = ∞, lim τ →0 τ which corresponds to the condition limτ →0 I2 (τ, which q is being replaced by m. 61), jointly with 0 < p < (N − 2)/2 and q ≥ q ∗∗ .

5. Assume that −∞ < λ < (N − 2)2 /4. 5). The following hold: ± (a) lim|x|→0 u(x)/Φ± λ (x) = ∞ if and only if lim sup|x|→0 u(x)/Φλ (x) = ∞; ± ± (b) lim|x|→0 u(x)/Φλ (x) = 0 if and only if lim inf |x|→0 u(x)/Φλ (x) = 0. 6. 5 hold with Ψ± instead of Φ± λ. Proof. (a) We need only prove that lim sup|x|→0 u(x)/Φ± λ (x) = ∞ implies (x) = ∞. Suppose by contradiction that that lim|x|→0 u(x)/Φ± λ l± := lim inf |x|→0 u(x) < ∞. Φ± λ (x) Then there exists a sequence (xn )n≥1 in RN which converges to zero such that ± limn→∞ u(xn )/Φ± λ (xn ) = l .

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 . 42 5. 7), we have limr→0 Φ− λ (r)/K(r) = 0.

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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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