Normalized solutions for the p-Laplacian equations with potentials and general nonlinearities

In this paper, we study a type of p -Laplacian equation − Δ p u + V ( x ) | u | p − 2 u + λ | u | p − 2 u = f ( u ) , x ∈ R N with the prescribed mass ( ∫ R N | u | p d x ) 1 p = c > 0 where N ≥ 3 , 1 < p < N , f ∈ C ( R , R ) , V ( x ) = | x | k ( k > 1 ) , Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian of u , λ ∈ R is Lagrange multiplier. We assume that f is odd and L p -supercritical. By some constrained minimization methods, we obtain a local minimizer u c which is a ground state for small mass c > 0 and we describe a mass collapse behavior of the minimizers as c → 0 + . Furthermore, under some assumptions of k , we study the asymptotic behavior of the corresponding Lagrange multiplier as c → 0 + . In addition, by a minimax approach based on the σ -homotopy stable family of the working space, we obtain a mountain pass type solution u ¯ c .