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We consider the existence of radial solutions for the quasilinear Schrödinger equation − △ u + V ( | x | ) u − κ u △ u 2 = | u | p − 2 u , x ∈ R N . where κ > 0 , N ≥ 3 , 2 < p < 2 2 ∗ and V ∈ C 1 ( R ) . Here we are interested in the case that 2 < p ≤ 4 since the existence of solutions for 4 < p < 2 2 ∗ can be easily obtained by a standard variational argument. Our method is based on the Pohožaev type identity. [ABSTRACT FROM AUTHOR]

In the present paper, we study the following Schrödinger–Kirchhoff quasilinear problem (Q) − ( a + b ∫ R N ∣ ∇ u ∣ 2 d x ) △ u − u △ ( u 2 ) + V ( x ) u = f ( x , u ) , x ∈ R N , where a > 0 , b ≥ 0 , N ≥ 3 and f ∈ C ( R N × R , R ) , which includes the Modified Nonlinear Schrödinger Equation. We prove the existence of infinitely many small energy solutions for equation (Q) by applying Clark’s Theorem to a perturbation functional. [ABSTRACT FROM AUTHOR]

Abstract: The main purpose of this paper is to unify and improve Theorems 3.1–3.3 in Wu (2012) [8] as well as some results in other literature such as Theorems 1–4 in Wu (2011) [9]. We present a new proof technique to conclude the existence of high energy solutions for the Kirchhoff-type system under some assumptions that are weaker than those in Wu (2012) [8]. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the periodic solutions of the following delay non-autonomous systemswhere r >0, f ∈ C(R 1 × R n , R n ) satisfies f(t + r, z)= f(t, z) for all z ∈ R n . Some multiplicity results of periodic solutions of (1) are obtained via variational methods. [Copyright &y& Elsevier]

Abstract: A discrete total variation calculus is presented in this paper. Using this calculus, we prove that the solution of the system of difference equations in Lee’s discrete mechanics preserves a symplectic structure in the space–time sense. Numerical experiments are also reported. [Copyright &y& Elsevier]

Abstract: We investigate the integrable deformation of the inhomogeneous Heisenberg ferromagnet model by using the prolongation structure theory. Through moving space curve in Euclidean space, the corresponding equivalent generalised inhomogeneous nonlinear Schrödinger equations are also presented. [Copyright &y& Elsevier]

Abstract: With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix of the eight-vertex solid-on-solid (SOS) model, we obtain the determinant representations of the scalar products of Bethe states for the open XYZ chain with non-diagonal boundary terms. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states. [Copyright &y& Elsevier]