• DOI:10.1515/anona-2017-0261
• Corpus ID: 85506223

Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

@article{Ghergu2018ExactBA,
  title={Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity},
  author={Marius Ghergu and Sunghan Kim and Henrik Shahgholian},
  journal={Advances in Nonlinear Analysis},
  year={2018},
  volume={8},
  pages={995 - 1003}
}

• M. Ghergu, Sunghan Kim, H. Shahgholian
• Published 12 April 2018
• Mathematics
• Advances in Nonlinear Analysis

Abstract We study the semilinear elliptic equation - Δ u = u α | log u | β   in B 1 ∖ { 0 } , -\Delta u=u^{\alpha}\lvert\log u|^{\beta}\quad\text{in }B_{1}\setminus\{0\}, where B 1 ⊂ ℝ n {B_{1}\subset{\mathbb{R}}^{n}} , with n ≥ 3 {n\geq 3} , n n - 2 < α < n + 2 n - 2 {\frac{n}{n-2}<\alpha<\frac{n+2}{n-2}} and - ∞ < β < ∞ {-\infty<\beta<\infty} . Our main result establishes that the nonnegative solution u ∈ C 2 ⁢ ( B 1 ∖ { 0 } ) {u\in C^{2}(B_{1}\setminus\{0\})} of the above equation either has a removable singularity at the origin or it behaves like u ⁢ ( x ) = A ⁢ ( 1 + o ⁢ ( 1 ) ) ⁢ | x | - 2 α - 1 ⁢ ( log ⁡ 1 | x | ) - β α - 1   as ⁢ x → 0 , u(x)=A(1+o(1))|x|^{-\frac{2}{\alpha-1}}\Bigl{(}\log\frac{1}{|x|}\Big{)}^{-% \frac{\beta}{\alpha-1}}\quad\text{as }x\rightarrow 0, with A = [ ( 2 α - 1 ) 1 - β ⁢ ( n - 2 - 2 α - 1 ) ] 1 α - 1 . {A=[(\frac{2}{\alpha-1})^{1-\beta}(n-2-\frac{2}{\alpha-1})]^{\frac{1}{\alpha-1% }}.}