On the bifurcation for fractional Laplace equations
Date Issued
2016-06-01
Author(s)
Dwivedi, Gaurav
Tyagi, Jagmohan
Verma, Ram Baran
Abstract
In this paper, we consider the bifurcation problem for fractional Laplace equation
(??)su=?u+f(?,x,u)in ?,u=0in Rn??,
where ??Rn,n>2s(0<s<1) is an open bounded subset with smooth boundary, (??)s stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue ?1 of the eigenvalue problem
(??)sv=?vin?,v=0inRn??,
and, conversely.
(??)su=?u+f(?,x,u)in ?,u=0in Rn??,
where ??Rn,n>2s(0<s<1) is an open bounded subset with smooth boundary, (??)s stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue ?1 of the eigenvalue problem
(??)sv=?vin?,v=0inRn??,
and, conversely.