Concentration phenomenon in the critical exponent problems on hyperbolic space. Issue 15 (18th November 2021) - Mukherjee, Tuhina
ABSTRACT: This article deals with the study of the following critical exponent problem 1 ( P λ, μ ) − Δ H n u + ( μ g ( x ) − λ ) u = u p − 1   i n   H n, u > 0   i n   H n, u ∈ H 1 ( H n ) where H n is the n -dimensional hyperbolic space, n ≥ 4, p = 2 n / ( n − 2 ) is the critical exponent, λ, μ > 0 are real parameters, Δ H n denotes the Laplace–Beltrami operator on H n and g ( x ) is a real valued potential function on H n . Using variational methods, we establish the existence of positive ground state solution to ( P λ, μ ) and study the convergence of these solutions when μ approaches + ∞ .