Communications on Pure & Applied Analysis
January 2021 , Volume 20 , Issue 1
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We prove the existence, positivity, simplicity, uniqueness up to nonnegative eigenfunctions, and isolation of the principle eigenvalue of a biharmonic system. We also provide the extension of our results to a related system.
Taking inspiration from a recent paper by Bergounioux et al., we study the Riemann-Liouville fractional Sobolev space
In this paper, we concern with the problem of limit cycle bifurcation for a class of piecewise smooth cubic systems. Using the first order Melnikov function we prove that at least thirteen limit cycles can be bifurcated from periodic solutions surrounding the center.
We consider the Cauchy problem of the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential. We prove that if any radial solution obeys the supreme of the kinetic energy over the maximal lifespan is below the kinetic energy of the ground state solution, then the solution exists globally in time and scatters in both time directions.
Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)
We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in
We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schrödinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.
In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherical obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.
We are concerned with the existence of blowing-up solutions to the following boundary value problem
We study eigenvalues and eigenfunctions of the Laplacian on the surfaces of four of the regular polyhedra: tetrahedron, octahedron, icosahedron and cube. We show two types of eigenfunctions: nonsingular ones that are smooth at vertices, lift to periodic functions on the plane and are expressible in terms of trigonometric polynomials; and singular ones that have none of these properties. We give numerical evidence for conjectured asymptotic estimates of the eigenvalue counting function. We describe an enlargement phenomenon for certain eigenfunctions on the octahedron that scales down eigenvalues by a factor of
We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,
In this paper, we investigate the approximations of stochastic
In this paper, we prove the existence and multiplicity of subharmonic bouncing motions for a Hill's type sublinear oscillator with an obstacle. Furthermore, we also consider the existence, multiplicity and dense distribution of symmetric periodic bouncing solutions when the weight function is even. Based on an appropriate coordinate transformation and the method of phase-plane analysis, we can study our main results via Poincar
In this paper we study the boundary regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations with nonzero boundary conditions. We construct global Hölder estimates for convex solutions to the problem and emphasize that the boundary regularity essentially depends on the convexity of the domain. The proof is based on a careful study of the concept of
We follow the idea of Wang [
We investigate the relationship between the sign of the discrete fractional sequential difference
Finite difference operators are widely used for the approximation of continuous ones. It is well known that the analysis of continuous differential operators may strongly depend on their dimensions. We will show that the finite difference operators generate the same algebra, regardless of their dimension.
We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work [
This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels
The main results in the paper are the weighted multipolar Hardy inequalities
The paper fits in the framework of Kolmogorov operators defined on smooth functions
perturbed by multipolar inverse square potentials, and related evolution problems. Necessary and sufficient conditions for the existence of exponentially bounded in time positive solutions to the associated initial value problem are based on weighted Hardy inequalities. For constants
The article aims to investigate the dynamic transitions of a toxin-producing phytoplankton zooplankton model with additional food in a 2D-rectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.
Nonlinear Schrödinger equations with an external magnetic field and a power nonlinearity with subcritical exponent
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