The first non-trivial Fučík curve for the sublinear Dirichlet Laplacian

Abstract

Let $Ω\subset\mathbb{R}^N$ be a bounded open connected set, and let $q\in(1,2)$. We consider the sublinear $q$-eigenvalue problem for the Dirichlet Laplacian $-Δu=λ\|u\|_{L^q(Ω)}^{2-q}|u|^{q-2}u$, $u\in H^1_0(Ω)\setminus\{0\}$, whose first eigenvalue $λ_1(Ω)$ is simple, with eigenfunctions of constant sign. The definition of a second eigenvalue is not unique. Following the approach introduced by Molle and Passaseo for the linear Laplacian, we construct the first non-trivial curve of the corresponding Fučík spectrum, and show that it has a unique diagonal point $(γ(Ω),γ(Ω))$. The Fučík-selected second value $λ_2^F(Ω)=2^{1-\frac{2}{q}}γ(Ω)$ is strictly larger than $λ_1(Ω)$, admits a two-phase variational characterisation, and satisfies a sharp Hong-Krahn-Szegő inequality: among sets of given measure it is minimised by the union of two equal balls. The selected solution at the diagonal point is a classical $(2,q)$-eigenfunction exactly when its nodal $L^q$-masses are balanced. We finally show that on balls in dimensions $N\geq2$ every Fučík-selected solution is foliated Schwarz symmetric.