scholarly journals Isolated singularity for semilinear elliptic equations

2015 ◽  
Vol 35 (7) ◽  
pp. 3239-3252 ◽  
Author(s):  
Lei Wei ◽  
◽  
Zhaosheng Feng ◽  
Keyword(s):  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Isolated Singularity ◽  
Semilinear Elliptic

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

2018 ◽  
Vol 8 (1) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Sunghan Kim ◽  
Henrik Shahgholian
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonnegative Solution ◽  
Removable Singularity ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Isolated Singularity ◽  
Semilinear Elliptic

Abstract We study the semilinear elliptic equation -\Delta u=u^{\alpha}\lvert\log u|^{\beta}\quad\text{in }B_{1}\setminus\{0\}, where {B_{1}\subset{\mathbb{R}}^{n}} , with {n\geq 3} , {\frac{n}{n-2}<\alpha<\frac{n+2}{n-2}} and {-\infty<\beta<\infty} . Our main result establishes that the nonnegative solution {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|^{-\frac{2}{\alpha-1}}\Bigl{(}\log\frac{1}{|x|}\Big{)}^{-% \frac{\beta}{\alpha-1}}\quad\text{as }x\rightarrow 0, with {A=[(\frac{2}{\alpha-1})^{1-\beta}(n-2-\frac{2}{\alpha-1})]^{\frac{1}{\alpha-1% }}.}

scholarly journals Nonexistence Results of Semilinear Elliptic Equations Coupled with the Chern-Simons Gauge Field

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Hyungjin Huh
Keyword(s):  
Gauge Field ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Semilinear Elliptic ◽  
Chern Simons

We discuss the nonexistence of nontrivial solutions for the Chern-Simons-Higgs and Chern-Simons-Schrödinger equations. The Derrick-Pohozaev type identities are derived to prove it.

scholarly journals Large solutions of semilinear elliptic equations with nonlinear gradient terms

1999 ◽  
Vol 22 (4) ◽  
pp. 869-883 ◽  
Author(s):  
Alan V. Lair ◽  
Aihua W. Wood
Keyword(s):  
Positive Solutions ◽  
Compact Support ◽  
Elliptic Equations ◽  
Nonnegative Function ◽  
Semilinear Elliptic Equations ◽  
Decay Condition ◽  
Large Solutions ◽  
Semilinear Elliptic ◽  
The Given

We show that large positive solutions exist for the equation(P±):Δu±|∇u|q=p(x)uγinΩ⫅RN(N≥3)for appropriate choices ofγ>1,q>0in which the domainΩis either bounded or equal toRN. The nonnegative functionpis continuous and may vanish on large parts ofΩ. IfΩ=RN, thenpmust satisfy a decay condition as|x|→∞. For(P+), the decay condition is simply∫0∞tϕ(t)dt<∞, whereϕ(t)=max|x|=tp(x). For(P−), we require thatt2+βϕ(t)be bounded above for some positiveβ. Furthermore, we show that the given conditions onγandpare nearly optimal for equation(P+)in that no large solutions exist if eitherγ≤1or the functionphas compact support inΩ.

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