CLASSIFICATION OF POSITIVE SMOOTH SOLUTIONS TO THIRD-ORDER PDES INVOLVING FRACTIONAL LAPLACIANS

In this paper, we are concerned with the third-order equations {(-Delta)(3/2) u = u(d+3/d-3), x is an element of R-d u is an element of C-3(R-d), u(x) > 0, x is an element of R-d, and {(-Delta)(3/2) u = (1/vertical bar x vertical bar(6) * vertical bar u vertical bar(2))u, x is an element of R-d, u is an element of C-3 (R-d), u(x) > 0, x is an element of R-d, d >= 7, with (H) over dot(3/2)-critical nonlinearity. By showing the equivalence between the PDEs and the corresponding integral equations and using results from Chen et al. (2006) and Dai et al. (2018), we prove that positive classical solutions a to the above equations are radially symmetric about some point x(0) is an element of R-d and derive the explicit forms for u.