Quasilinear problems with singularities
This paper gives an existence and uniqueness result for: \[ (*)\quad - div(| \nabla u|^{p-2}\nabla u)=\mu \quad in\quad {\mathbb{R}}^ N;\quad u\to 0\quad at\quad \inf inity \] where \(\mu\) is a sum of Dirac masses; \(1<p<\infty\), \(N\geq 2\), and one must impose \((\mu,1)=0\) if \(p\geq N\) for u to tend to 0. For the proof one has to distinguish three cases: if \(p>N\), one uses a variational method; if \(p<N\), one establishes bounds on approximate problems with right-hand side in \(L^ 1\), by a refinement of known symmetrization methods and a regularity theorem; if \(p=N\), these methods break down and one solves instead of (*) an equivalent problem on the N-sphere, and introduces a symmetrization technique for functions of variable sign on the sphere. Two other results, of independent interest, are also proved in this work. The first is a regularity theorem for some nonlinear systems on differential forms with non ''pure power'' nonlinearity. This result is further improved thanks to Hodge duality; thus, one obtains the interior regularity of p-harmonic functions (for \(1<p<2)\) as a consequence of the Hölder continuity of (N-1)-forms solving \(d\omega =0\), \(\delta (| \omega |^{p'-2}\omega)=0\), \(p'=p/(p-1)>2\). The second result is the construction of the perimeter of a general subset of a compact orientable Riemannian manifold, generalizing de Giorig's perimeter of subsets of \(R^ N\). This construction makes essential use of the properties of the heat kernel on differential forms.
scientific article; zbMATH DE number 3957738
scientific article