scholarly journals An existence theorem for positive solutions of semilinear elliptic equations

1986 ◽  
Vol 95 (3) ◽  
pp. 211-216 ◽  
Author(s):  
Joel Smoller ◽  
Arthur Wasserman
Keyword(s):  
Existence Theorem ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic

scholarly journals On positive solutions of some semilinear elliptic equations

1991 ◽  
Vol 50 (3) ◽  
pp. 343-355 ◽  
Author(s):  
Shin-Hwa Wang ◽  
Nicholas D. Kazarinoff
Keyword(s):  
Lower Bound ◽  
Existence Theorem ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Number Of Solutions ◽  
Nonexistence Theorem ◽  
Semilinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Bound Theorem

AbstractThe existence of positive solutions of some semilinear elliptic equations of the form −Δu = λf(u) is studied. The major results are a nonexistence theorem which gives a λ* = λ*(f,Ω) > 0 below which no positive solutions exist and a lower bound theorem for umax for Ω a ball. As a corollary of the nonexistence theorem that describes the dependence of the number of solutions on λ, two other nonexistence theorems, and an existence theorem are also proved.

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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