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

In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrodinger equation with mixed dispersion: delta Delta(2)u - Delta u + u = vertical bar u vertical bar(2)sigma u, u is an element of H-2(R-N), where delta > 0 is sufficiently small, 0 < sigma < 2/(N-2)(+), 2/(N-2)(+) = 2/N-2 for N >= 3 and 2/(N-2)(+) = +infinity 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 Schrodinger 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| ->+infinity, P(r) = 1 + a(1)/r(m1) + O(1/r(m1)+theta(1)), Q(r) = 1 + a(2)/r(m2) + O(1/r(m2)+theta(2)), where a1, a2 is an element of R, m(1), m(2) > 1,theta(1), theta 2(1) > 0. We use the Lyapunov- Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrodinger equations in RN. Calc Var. 2010;37:423-439] to construct infinitely many nonradial positive and signchanging solutions with arbitrary large energy for the following equation: delta Delta(2)u - Delta u + P(x)u = Q(x)vertical bar u vertical bar(2)sigma u, u is an element of H-2(R-N).