Payne nodal set conjecture for the fractional p-Laplacian in Steiner symmetric domains
arxiv(2024)
摘要
Let u be either a second eigenfunction of the fractional p-Laplacian or a
least energy nodal solution of the equation (-Δ)^s_p u = f(u) with
superhomogeneous and subcritical nonlinearity f, in a bounded open set
Ω and under the nonlocal zero Dirichlet conditions. Assuming only that
Ω is Steiner symmetric, we show that the supports of positive and
negative parts of u touch ∂Ω. As a consequence, the nodal set
of u has the same property whenever Ω is connected. The proof is based
on the analysis of equality cases in certain polarization inequalities
involving positive and negative parts of u, and on alternative
characterizations of second eigenfunctions and least energy nodal solutions.
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