Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials. (February 2022) - Diebou Yomgne, Gael
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, C l o c 2, 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 ) 2 m − 1 2 v = K ( x ) | v | σ − 1 v + H ( x ) v + g in R n, m ∈ N for 2 ≤ 2 m < n + 1 is solvable in a suitable Morrey–Lorentz space and admits positive solutions whenever σ > n / ( n − 2 m + 1 ) . We also prove that no positive solution in the latter framework exists if σ ≤ n / ( n − 2 m + 1 ) .