We concern the following discrete nonlinear Schrödinger equation with steep potential $$\begin{cases} L u_n + \lambda v_n u_n - \omega u_n = f(u_n), \quad n \in \mathbb{Z}, \\ \lim\limits_{|n| \to \infty} u_n = 0,\end{cases}$$ where λ is a positive parameter, $L$ is a Jacobi operator, $ω$ is a real number. The nonlinear term $f ∈C(\mathbb{R},\mathbb{R})$ is assumed to satisfy superlinear conditions and discrete potential $V=\{v_n\}$ is a sequence of real numbers. Under suitable assumptions, we establish the existence of ground state solutions and ground state sign-changing solutions using variational methods. Furthermore, we investigate their concentration behaviors as $λ→∞.$ Lastly, motivated by Y. H. Ding and K. Tanaka in [1], we consider the existence of multi-bump solutions of this equation.
In this work, we study the regularity of the Cauchy problem for the free transport equation, and by using the inverse Wigner transformation, we reduce this problem to the Cauchy problem of a class of linear homogeneous hyperbolic Schro & uml;dinger equation. We prove firstly the analytical smoothing effect of Cauchy problem for Schro & uml;dinger type equation if the initial datum is exponential decay. Finally we prove the directional propagation of the exponential decay and also analytic regularity for free transport equation.
We address the problem given by the following partial differential equation: some semi-Linear parabolic equations with uniformly elliptic non-local operators in Half-Space. Initially, we establish a generalized weighted average inequality and a maximum principle in unbounded domains, which are crucial for the sliding method. Then, we employ sliding to demonstrate the monotonicity of bounded positive solutions. In this paper, we will remove the monotonicity assumption of the kernel function a(x) by using the sliding method. The techniques employed in the process of this method have applications to other problems related to uniformly elliptic operators.
In this paper, we consider an eigenvalue problem for a class of nonlinear operators containing p(& centerdot;)-Laplacian and mean curvature operator with mixed boundary conditions. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show that the eigenvalue problem has infinitely many eigenpairs by using the celebrated Ljusternik-Schnirelmann principle of the calculus of variation. Moreover, in a variable exponent Sobolev space, there are two cases where the infimum of all eigenvalues is equal to zero and is positive.
In this paper, we consider the structure of the L2-spectrum of the sub-Laplacian on 2-step stratified Lie groups by using the theory of unitary irreducible representations and Hermit functions. We extend the results for Heisenberg group and H-type Lie group to more general 2-step stratified Lie groups without the Moore-Wolf condition.
