Nonlocal problem with critical exponential nonlinearity of convolution type: A non-resonant case

2024/08/01 by Kanungo, Suman, Mishra, Pawan Kumar
#35J20 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35J15

paper · doi:10.48550/arxiv.2408.00654

Abstract

In this paper, we study the following class of weighted Choquard equations -Δu =λu + (∫_Ω(Q(|y|)F(u(y)))/(|x-y|^μ)dy) Q(|x|)f(u) ~~\textrmin~~ Ω~~ and~~ u=0~~ \textrmon~~ ∂ Ω, where Ω⊂ ℝ² is a bounded domain with smooth boundary, μ∈ (0,2) and λ>0 is a parameter. We assume that f is a real valued continuous function satisfying critical exponential growth in the Trudinger-Moser sense, and F is the primitive of f. Let Q be a positive real valued continuous weight, which can be singular at zero. Our main goal is to prove the existence of a nontrivial solution for all parameter values except the resonant case, i.e., when λ coincides with any of the eigenvalues of the operator (-Δ, H¹₀(Ω)).

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