Existence and Decay Estimates of Minimizers for the p -Laplacian Schrödinger Energy Functional with Hardy Potential and Sobolev Critical Growth

In this paper, we discuss the following nonlinear p -Laplacian problems with a Hardy term and Sobolev critical growth: { − Δ p u − μ | x | p | u | p − 2 u + λ | u | p − 2 u = | u | p * − 2 u + κ | u | q − 2 u , x ∈ R N ∖ { 0 } , ∫ R N | u | p d x = c , where N ≥ 2 , 1 < p < N , c , κ > 0 , p * := p + p 2 / N < q < p * := N p / ( N − p ) , μ < μ ¯ := ( ( N − p ) / p ) p and λ ∈ R is an unknown Lagrange multiplier. The central focus of this paper is to demonstrate the existence of normalized ground state solutions for this problem, which possesses an exponential decay property at infinity. Moreover, for any sequence { μ n } ⊂ ( 0 , + ∞ ) satisfying μ n → 0 + , we prove that the corresponding sequence of these ground state solutions converges to a ground state solution of { − Δ p u + λ | u | p − 2 u = | u | p * − 2 u + κ | u | q − 2 u , x ∈ R N ∖ { 0 } , ∫ R N | u | p d x = c .