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

In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: δΔ2u−Δu+u=|u|2σu,u∈H2(RN),where δ>0 is sufficiently small, 0<σ<2(N−2)+, 2(N−2)+=2N−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+a1rm1+O(1rm1+θ1),Q(r)=1+a2rm2+O(1rm2+θ2),where a1,a2∈R, m1,m2>1, θ1,θ2>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: δΔ2u−Δu+P(x)u=Q(x)|u|2σu,u∈H2(RN).