A Fourier approach for nonlinear equations with singular data

For 0 < m < n, p a positive integer and p > n/(n ? m), we study the inhomogeneous equation L _u +u ^p + V (x)u + f(x) = 0 in ? ⁿ with singular data f and V. The symbol σ of the operator L is bounded from below by |ξ| ^m . Examples of L are Laplacian, biharmonic and fractional order operators. Here f and V can have infinite singular points, change sign, oscillate at infinity, and be measures. Also, f and V can blow up on an unbounded (n?1)-manifold. The solution u can change sign, be nonradial and singular. If σ, f and V are radial, then u is radial. The assumptions on f and V are in terms of their Fourier transforms and we provide some examples.