Electron. J. Differential Equations, Vol. 2021 (2021), No. 52, pp. 1-24.

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

Pablo Alvarez-Caudevilla, Eduardo Colorado, Alejandro Ortega

Abstract:
This article concerns the existence of positive solutions for the second order equation involving a nonlocal term

under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter γ>0, and up to the critical value of the exponent p, i.e. when 1<p<q; 2*-1, where 2*=(2N)/(N-2) is the critical Sobolev exponent. For p=2*-1, this leads us to a Brezis-Nirenberg type problem, cf. [5], but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant, that ensures the existence of positive solution, going from dimensions N≥4 in the classical Brezis-Nirenberg problem, to dimensions N≥7 for this nonlocal problem.

Submitted November 3, 2020. Published June 14, 2021.
Math Subject Classifications: 35G20, 35A15, 35J50, 35B38, 35J91.
Key Words: Critical point; concentration compactness principle; mountain pass theorem.
DOI: https://doi.org/10.58997/ejde.2021.52

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Pablo Álvarez-Caudevilla
Departamento de Matemáticas
Universidad Carlos III de Madrid
Av. Universidad 30, 28911, Leganés, Madrid, Spain
email: pacaudev@math.uc3m.es
Eduardo Colorado
Departamento de Matemáticas
Universidad Carlos III de Madrid
Av. Universidad 30, 28911, Leganés, Madrid, Spain
email: ecolorad@math.uc3m.es
Alejandro Ortega
Departamento de Matemáticas
Universidad Carlos III de Madrid
Av. Universidad 30, 28911, Leganés, Madrid, Spain
email: alortega@math.uc3m.es

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