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Existence of Infinitely Many Solutions for Singular Semilinear Problems on Exterior Domains (open access)

Existence of Infinitely Many Solutions for Singular Semilinear Problems on Exterior Domains

Article proving the existence of infinitely many radial solutions of โˆ†๐“Š+๐พ(๐“‡) ฦ’ (๐“Š) = 0 on the exterior of the ball of radius R > 0, BR, centered at the origin in โ„แดบ with u = 0 on โˆ‚BR and lim/rโ†’โˆž ๐“Š(๐“‡) = 0 where N > 2, f is odd with f < 0 on (0, ฮฒ), f > 0 on (ฮฒ, โˆž), f is superlinear for large u, ฦ’(๐“Š) โˆผ โˆ’1/(|๐“Š|๐˜ฒโปยน๐“Š) with 0 < q < 1 for small u, and 0 < ๐พ(๐“‡) โ‰ค ๐พโ‚/rโˆ with ๐‘ + q(๐‘ โˆ’ 2) < โˆ < 2(๐‘โˆ’ 1) for large r.
Date: March 9, 2018
Creator: Iaia, Joseph A.
System: The UNT Digital Library
Loitering at the hilltop on exterior domains (open access)

Loitering at the hilltop on exterior domains

In this article, the author proves the existence of an infinite number of radial solutions of ฮ”u+f(u)=0 on the exterior of the ball of radius R>0 centered at the origin and f is odd with f<0 on (0,ฮฒ), f>0 on (ฮฒ,ฮด), and fโ‰ก0 for u>ฮด. The primitive F(u)=โˆซu0f(t)dt has a "hilltop" at u=ฮด which allows one to use the shooting method and ODE techniques to prove the existence of solutions.
Date: November 23, 2015
Creator: Iaia, Joseph A.
System: The UNT Digital Library