Acta mathematica scientia, Series B >
NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SEMI-LINEAR EQUATION INVOLVING THE FRACTIONAL LAPLACIAN IN RN
Received date: 2015-05-13
Revised date: 2015-09-09
Online published: 2016-06-25
In this paper, we consider the semilinear equation involving the fractional Lapla-cian in the Euclidian space Rn:
(-Δ)α/2u(x)=f(xn)up(x), x∈Rn (0.1)
in the subcritical case with 1< p< (n+α)/(n-α). Instead of carrying out direct investigations on pseudo-differential equation (0.1), we first seek its equivalent form in an integral equation as below:
u(x)=∫Rn G∞(x, y) f(yn)up(y)dy, (0.2)
where G∞(x, y) is the Green's function associated with the fractional Laplacian in Rn. Em-ploying the method of moving planes in integral forms, we are able to derive the nonexistence of positive solutions for (0.2) in the subcritical case. Thanks to the equivalence, same con-clusion is true for (0.1).
Yan LI . NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SEMI-LINEAR EQUATION INVOLVING THE FRACTIONAL LAPLACIAN IN RN[J]. Acta mathematica scientia, Series B, 2016 , 36(3) : 666 -682 . DOI: 10.1016/S0252-9602(16)30030-3
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