scholarly journals Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains

1990 ◽  
Vol 86 (2) ◽  
pp. 367-391 ◽  
Author(s):  
Song-Sun Lin
Keyword(s):  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Radial Solutions ◽  
Semilinear Elliptic ◽  
Positive Radial Solutions ◽  
Annular Domains

scholarly journals Positive and oscillatory radial solutions of semilinear elliptic equations

1997 ◽  
Vol 10 (1) ◽  
pp. 95-108 ◽  
Author(s):  
Shaohua Chen ◽  
William R. Derrick ◽  
Joseph A. Cima
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Elliptic Equations ◽  
Nonlinear Partial Differential Equation ◽  
Semilinear Elliptic Equations ◽  
Radial Solutions ◽  
Partial Differential ◽  
Semilinear Elliptic ◽  
Positive Radial Solutions

We prove that the nonlinear partial differential equation Δu+f(u)+g(|x|,u)=0, in  ℝn,n≥3, with u(0)>0, where f and g are continuous, f(u)>0 and g(|x|,u)>0 for u>0, and limu→0+f(u)uq=B>0, for 1<q<n/(n−2), has no positive or eventually positive radial solutions. For g(|x|,u)≡0, when n/(n−2)≤q<(n+2)/(n−2) the same conclusion holds provided 2F(u)≥(1−2/n)uf(u), where F(u)=∫0uf(s)ds. We also discuss the behavior of the radial solutions for f(u)=u3+u5 and f(u)=u4+u5 in ℝ3 when g(|x|,u)≡0.

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