Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials

Abstract

We establish the boundedness properties of the Neumann extension operator in half-space in the setting of Morrey–Lorentz spaces. As a by-product we derive estimates on the restriction operator in block spaces. Direct application to solvability of a fourth order nonlinear equation related to higher order boundary conformally invariant problem is considered. By employing a nonvariational approach, we obtain a unique solution under suitable smallness conditions on the boundary data and potentials prescribed in Morrey–Lorentz spaces which allow for singular functions. Moreover, these solutions are shown to be C∞ in the interior, Cloc2,μ(R+n+1¯), μ∈(0,1) in some special case and satisfy interesting qualitative properties including positivity. The results are extended to a larger class of problems involving the polyharmonic operator. In particular, the higher order nonlocal equation (−Δ)2m−12v=K(x)|v|σ−1v+H(x)v+ginRn,m∈Nfor 2≤2m<n+1 is solvable in a suitable Morrey–Lorentz space and admits positive solutions whenever σ>n/(n−2m+1). We also prove that no positive solution in the latter framework exists if σ≤n/(n−2m+1).