This paper is devoted to the multiplicity of normalized solutions for a class of Schrödinger equations with competing Van der Waals potentials, which arise from a model describing non-relativistic bosonic atoms and molecules. By developing an approach based on the direct minimization of the energy functional on a newly constructed constrained manifold, we provide more accurate sufficient conditions for existence of two solutions. More importantly, the upper bounds estimates of the interaction parameter and energy level can be explicitly derived. Our results complement the existing theory for both nonlocal and local Schrödinger equations, and provide a new and simpler approach.
In this paper, we shall investigate the long-time asymptotic behavior of the wave equation with a nonlocal strong damping, an nonlocal anti-damping and a subcritical nonlinearity. We shall establish the global well-posedness by extending the semigroup-approximation method introduced by Lasiecka et al. in [14]. Under the assumption that f(u) is of subcritical growth, the existence of the global attractor is then obtained via the Condition (C) method.
Consider the following nonlinear Kirchhoff type problem - (a+b∫ _ℝ^N|∇ u|^2 )Δ u+V(x)u=|u|^p-1u, in ℝ^N, where a,b>0 , N≥ 3 , 10 , ξ >0, m>1 , and μ >0 are constants. Moreover, the energy of these solutions can be made arbitrarily large.
We present a primal-dual proximal splitting framework for solving a broad class of convex semidefinite optimization problems with distributionally robust constraints involving piecewise affine functions, spectrahedral support sets, and Wasserstein ambiguity, which arise in a broad range of applications. We develop a semidefinite programming (SDP)-based formula for computing the projection onto a spectrahedron enabling the integration of SDP techniques within a primal-dual splitting framework for solving these problems via finite-dimensional convex semidefinite optimization reformulations. By reformulating the distributionally robust constraints into a system of linear matrix inequalities, we express the original distributionally robust problem as a finite-dimensional convex problem which results in convex composite unconstrained optimization problems suitable for primal-dual projection algorithms. Finally, we demonstrate the computational tractability of the proposed framework through numerical experiments on two representative models: mean-risk portfolio optimization with transaction costs in finance, and binary classification in machine learning. The results reveal that the primal-dual algorithm effectively solves the models of moderate scale.
In this paper, we consider a transmission model of coupled wave equation and Kirchhoff-type wave equation in different domains through a common boundary, in which viscoelastic material is pasted into the edge of the Kirchhoff-type wave equation. When the kernel function satisfies a second-order nonlinear ordinary differential equation, applying inverse Volterra’s operator and the Riemannian geometry method, we establish the general decay result which is given by solutions of a first order nonlinear, dissipative ODE under some conditions on the coefficient matrix, and which is not necessarily decaying in a polynomial or exponential fashion.
