Concave-Convex critical problems for the spectral fractional Laplacian with mixed boundary conditions

2022/01/31 by Alejandro Ortega, Ortega, Alejandro · 2 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2201.13154

Abstract

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \ (-Δ)^su=λu^q+u^{2_s^*-1}, u · gt;0\quadin Ω,
\mkern+51mu u=0\quadon Σ_D
\mkern+36mu (∂ u)/(∂ ν)=0\quadon Σ_N . where Ω⊂ℝ^N is a smooth bounded domain, (1)/(2)0, ν is the outwards normal to ∂Ω, Σ_D, Σ_N are smooth (N-1)-dimensional submanifolds of ∂Ω such that Σ_D∪Σ_N=∂Ω, Σ_D∩Σ_N=∅, and Σ_D∩Σ_N=Γ is a smooth (N-2)-dimensional submanifold of ∂Ω.\newline In particular, we will prove that, for the sublinear case 00. We will also prove that solutions are bounded.

Cited by

• Fractional critical systems with mixed boundary conditions
• Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations

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