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In this paper, we investigate the structure and stability of the isotropic-nematic interface in 1-D. In the absence of the anisotropic energy, the uniaxial solution is the only global minimizer. In the presence of the anisotropic energy, the uniaxial solution with the homeotropic anchoring is stable for $$L_2<0$$ and unstable for $$L_2>0$$ . We also present many interesting open questions, some of which are related to De Giorgi conjecture. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the following non-local superlinear elliptic problem (-Δ)su=|u|p-σϕ1inΩ,u=0inRN\Ω,u>0inΩ,where (-Δ)s is the fractional Laplace operator, Ω⊂RN is an open domain with Lipschitz boundary, σ>0, p∈(1,2s∗-1) with 2s∗=2NN-2s and ϕ1 is the first positive eigenfunction of the fractional Laplacian with Dirichlet boundary condition. We prove the non-local version of a conjecture due to Lazer and McKenna (Proc R Soc Edinb Sect A 95:275-283, 1983) by constructing solutions with sharp peaks near local maximum points of ϕ1. [ABSTRACT FROM AUTHOR]