A Fourier approach for nonlinear equations with singular data
For \(0<m<n\), \(p\) a positive integer and \(p>n/(n-m)\), the authors study the inhomogeneous equation \(Lu+u^p+V(x)u+f(x)=0\) in \({\mathbb R}^n\) with singular data \(f\) and \(V\). The symbol \(\sigma\) of the operator \(L\) is bounded from below by \(|\xi|^m\). 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 \(\sigma\), \(f\) and \(V\) are radial, then \(u\) is radial. The assumptions of \(f\) and \(V\) are in terms of their Fourier transform, and some examples are provided. Usually, the Fourier transform is a powerful tool to solve linear partial differential equations, because the resulting algebraic equation can be solved for \(\hat{u}\). And by inversion, \({\hat{u}}\) turns out to be a solution of the original equation. In this paper, the authors successfully carry out a resembling strategy for a nonlinear equation and the corresponding nonalgebraic equation in Fourier variables.
scientific article; zbMATH DE number 6176903
scientific article