Infinitely many solutions for nonlinear fourth-order Schrödinger equations with mixed dispersion

Abstract

ABSTRACT In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: δ Δ 2 u − Δu + u = | u | 2 σ u , u ∈ H 2 ( R N ) , where δ > 0 is sufficiently small, 0 < σ < 2 ( N − 2 ) + , 2 ( N − 2 ) + = 2 N − 2 for N ≥ 3 and 2 ( N − 2 ) + = + ∞ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose P ( x ) and Q ( x ) are two positive, radial and continuous functions satisfying that as r = | x | → + ∞ , P ( r ) = 1 + a 1 r m 1 + O ( 1 r m 1 + θ 1 ) , Q ( r ) = 1 + a 2 r m 2 + O ( 1 r m 2 + θ 2 ) , where a 1 , a 2 ∈ R , m 1 , m 2 > 1 , θ 1 , θ 2 > 0 . We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: δ Δ 2 u − Δu + P ( x ) u = Q ( x ) | u | 2 σ u , u ∈ H 2 ( R N ) .