Positive solutions of nonlinear Schrödinger equation with peaks on a Clifford torus

We prove the existence of large energy positive solutions for a stationary nonlinear Schrödinger equation Δ u − V ( x ) u + u p = 0inR Nwith peaks on a Clifford type torus. HereV ( x ) = V ( r 1 , r 2 , ⋯ , r s )=1 + 1 ( a 1 r 1 m + a 2 r 2 m + a 3 r 3 m + ⋯ + a s r s m )+ O 1 ( a 1 r 1 m + a 2 r 2 m + a 3 r 3 m + ⋯ + a s r s m ) 1 + τwhereR N = R N 1 × R N 2 × ⋯ × R N s , withN i ≥ 2 for all i = 1 , 2 , ⋯ , s ,m > 1 , τ > 0 , r i = | x i | . Each r i is a function r , ϕ 1 , ⋯ , ϕ i − 1and is defined by the generalized notion of spherical coordinates. The solutions are obtained by a max ( r , ϕ 1 , ⋯ , ϕ s − 1 )or amax r min ( ϕ 1 , ⋯ , ϕ s − 1 )process.