Existence of positive solutions for a Brezis--Nirenberg type problem involving an inverse operator

2019/03/11 by Pablo Álvarez‐Caudevilla, Álvarez-Caudevilla, Pablo, Eduardo Colorado +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1903.04345

Abstract

This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, (-Δ)² u=λu+ (-Δ)|u|^p-1u, in a bounded domain Ω⊂ℝ^N, N≥ 7, and assuming homogeneous Navier boundary conditions. In particular, we study a second order equation involving a nonlocal term of the form, -Δu=λ(-Δ)^-1 u+|u|^p-1u, under Dirichlet boundary conditions and we prove the existence of positive solutions depending on the positive real parameter λ>0, up to the critical value of the exponent p, i.e., when 1

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