Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains.
Let \(M\) be a smooth connected compact boundaryless manifold of real dimension equipped with a Riemann metric tensor. Let us consider the Laplace-Beltrami operator on \(M\): \[ \Delta u \equiv \text{ div(grad } u) , \] and the semilinear elliptic PDE: \[ \Delta u - N(x,u)= F(x), \] associated either to Dirichlet or to Neumann boundary conditions, in a Lipschitz domain \(\Omega \subset M.\) In the case of Dirichlet boundary conditions the authors consider \(u \in L^p_{s+1/p}(\Omega),\) \(F \in L^p_{s+1/p-2}(\Omega),\) \(0<s<1,\) \(1<p<\infty\) and either \(N(x,u)\) has sublinear growth in \(u,\) or, if \(N \in C^1,\) the PDE become \[ \Delta u- a(x,u)\cdot u = f \] where the term \(a\) has an admissible polynomial growth. The case of Neumann boundary condition is similar. The main goal is to study the maximal range of indices \(s, p\) for which the semilinear elliptic PDE is solvable in the context of Lipschitz domains on the Sobolev-Besov scales. The obtained results are sharp in the sense that they are reduced to an optimal linear theory in the absence of nonlinearities.
scientific article; zbMATH DE number 1876875