4 Effect of the Hardy–Leray potential in the solvability of semilinear elliptic equations

2021 ◽  
pp. 45-70
Keyword(s):  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic

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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