Boundary Value Problems

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List of Papers (Total 1,627)

The continuum branch of positive solutions for discrete simply supported beam equation with local linear growth condition

In this paper, we obtain the global structure of positive solutions for nonlinear discrete simply supported beam equation Δ 4 u ( t − 2 ) = λ f ( t , u ( t ) ) , t ∈ T , u ( 1 ) = u ( T + 1 ) = Δ 2 u ( 0 ) = Δ 2 u ( T ) = 0 , $$\begin{aligned}& \Delta ^{4}u(t-2)= \lambda f\bigl(t,u(t)\bigr),\quad t\in \mathbb{T}, \\& u(1)=u(T+1)=\Delta ^{2}u(0)=\Delta ^{2}u(T)=0, \end{aligned...

A free boundary problem for an attraction–repulsion chemotaxis system

In this paper we study an attraction–repulsion chemotaxis system with a free boundary in one space dimension. First, under some conditions, we investigate existence, uniqueness and uniform estimates of the global solution. Next, we prove a spreading–vanishing dichotomy for this model. In the vanishing case, the species fail to establish and die out in the long run. In the...

On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions

We give a new approach for the estimations of the eigenvalues of non-self-adjoint Sturm–Liouville operators with periodic and antiperiodic boundary conditions. Moreover, we give error estimations, and finally we present some numerical examples.

Homoclinic solutions for second-order Hamiltonian systems with periodic potential

In this paper, we study the second-order Hamiltonian systems u ¨ − L ( t ) u + ∇ W ( t , u ) = 0 , t ∈ R , $$ \ddot{u}-L(t)u+\nabla W(t,u)=0,\quad t\in \mathbb{R}, $$ where L ∈ C ( R , R N × N ) $L\in C(\mathbb{R},\mathbb{R}^{N\times N})$ is a T-periodic and positive definite matrix for all t ∈ R $t\in \mathbb{R}$ and W is superquadratic but does not satisfy the usual Ambrosetti...

Inhomogeneous boundary value problems for steady compressible magnetohydrodynamic equations

This paper will be concerned with the compressible perturbation to steady magnetohydrodynamic equations near a uniform flow. In particular, the velocity of the basic flow under consideration can be any non-zero constant. We prove that there exists a stationary strong solution around a given basic flow with inhomogeneous boundary condition.

Multiplicity results for the Kirchhoff type equation via critical groups

In this paper, we will compute critical groups at zero for the Kirchhoff type equation using the property that critical groups are invariant under homotopies preserving isolatedness of critical points. Using this results, we can get more nontrivial solutions when the functional of this equation is coercive.

Fast homoclinic solutions for damped vibration problems with superquadratic potentials

In this paper we investigate the existence and multiplicity of homoclinic solutions for the following damped vibration problem: DS u ¨ + q ( t ) u ˙ − L ( t ) u + W u ( t , u ) = 0 , $$ \ddot{u}+q(t) \dot{u}-L(t)u+W_{u}(t,u)=0, $$ where q : R → R $q:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function, L ∈ C ( R , R n 2 ) $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is a...

Well-posedness of a higher-order Schrödinger–Poisson–Slater system

In this paper, we show the global well-posedness of a higher-order nonlinear Schrödinger equation. Specifically, we consider a system of infinitely many coupled higher-order Schrödinger–Poisson–Slater equations with a self-consistent Coulomb potential. We prove the existence and uniqueness global in time of solutions in L 2 ( R 3 ) $L^{2}( \mathbb{R}^{3})$ and in the energy space.

Superlinear Kirchhoff-type problems of the fractional p -Laplacian without the (AR) condition

In this paper, we study the following superlinear p-Kirchhoff-type equation: { M ( ∫ R 2 N | u ( x ) − u ( y ) | p | x − y | N + p s d x d y ) ( − △ ) p s u ( x ) − λ | u | p − 2 u = g ( x , u ) in  Ω , u = 0 in  R N ∖ Ω , $$\begin{aligned} \textstyle\begin{cases} \mathcal{M} (\int_{\mathbb{R}^{2N}}\frac { \vert u(x)-u(y) \vert ^{p}}{ \vert x-y \vert ^{N+ps}}\,dx\,dy )(-\triangle...

Topological properties of C 0 $C^{0}$ -solution set for impulsive evolution inclusions

In this paper, we study the topological properties to a C 0 $C^{0}$ -solution set of impulsive evolution inclusions. The definition of C 0 $C^{0}$ -solutions for impulsive functional evolution inclusions is introduced. The R δ $R_{\delta}$ -property of C 0 $C^{0}$ -solution set is studied for compact as well as noncompact semigroups on compact intervals. Applying the inverse...

Dynamics of a predator–prey system with three species

This paper is concerned with the dynamics of a predator–prey system with three species. When the domain is bounded, the global stability of positive steady state is established by contracting rectangles. When the domain is R $\mathbb{R}$ , we study the traveling wave solutions implying that one predator and one prey invade the habitat of another prey. More precisely, the...

Global existence of solutions for a class of thermoelastic plate systems

This paper is concerned with the initial-boundary value problem for a class of thermoelastic plate systems. Under some appropriate assumptions, the global existence of solutions is obtained.

Existence of stable standing waves for the Schrödinger–Choquard equation

In this paper, by variational methods and the profile decomposition of bounded sequences in H 1 $H^{1}$ we study the existence of stable standing waves for the Schrödinger–Choquard equation with an L 2 $L^{2}$ -critical nonlinearity. Our results extend some earlier results.

Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level

In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation u t − △ u t − △ u − div ( | ∇ u | 2 q ∇ u ) = u p $$ u_{t}-\triangle u_{t}-\triangle u-\operatorname{div}\bigl(| \nabla u|^{2q}\nabla u\bigr)=u^{p} $$ which was studied by Peng et al. (Appl. Math. Lett. 56:17–22, 2016), where the blow-up phenomena and the lifespan...

General decay and blow-up of solutions for a nonlinear viscoelastic wave equation with strong damping

This article is concerned with the decay and blow-up properties of a nonlinear viscoelastic wave equation with strong damping. We first show a local existence theorem. Then, we prove the global existence of solutions and establish a general decay rate estimate. Finally, we show the finite time blow-up result for some solutions with negative initial energy and positive initial...

Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary conditions

In this paper, we consider a class of fractional Langevin equations with integral and anti-periodic boundary conditions. By using some fixed point theorems and the Leray–Schauder degree theory, several new existence results of solutions are obtained.

Existence of solution for integral boundary value problems of fractional differential equations

In this paper, we discuss the existence of positive solutions of fractional differential equations on the infinite interval ( 0 , + ∞ ) $(0,+\infty)$ . The positive solution of fractional differential equations is gained by using the properties of the Green’s function, Leray–Schauder’s fixed point theorems, and Guo–Krasnosel’skii’s fixed point theorem. As an application, two...

On a p ( x ) $p(x)$ -biharmonic problem with Navier boundary condition

In this paper, we study a p ( x ) $p(x)$ -biharmonic equation with Navier boundary condition { Δ p ( x ) 2 u + a ( x ) | u | p ( x ) − 2 u = λ f ( x , u ) + μ g ( x , u ) in  Ω , u = Δ u = 0 on  ∂ Ω . $$ \textstyle\begin{cases} \Delta^{2}_{p(x)}u+a(x)|u|^{p(x)-2}u= \lambda f(x,u)+\mu g(x,u)\quad \text{in } \Omega, \\ u=\Delta u=0 \quad \text{on } \partial\Omega. \end{cases...

Fixed point theorems for a class of nonlinear sum-type operators and application in a fractional differential equation

In this paper, we consider the fixed point for a class of nonlinear sum-type operators ‘ A + B + C $A+B+C$ ’ on an ordered Banach space, where A, B are two mixed monotone operators, C is an increasing operator. Without assuming the existence of upper-lower solutions or compactness or continuity conditions, we prove the unique existence of a positive fixed point and also construct...

A diffusive stage-structured model with a free boundary

In this paper we mainly consider a free boundary problem for a single-species model with stage structure in a radially symmetric setting. In our model, the individuals of a new or invasive species are classified as belonging either to the immature or to the mature cases. We firstly study the asymptotic behavior of the solution to the corresponding initial problem, then obtain a...

A global nonexistence of solutions for a quasilinear viscoelastic wave equation with acoustic boundary conditions

In this paper, we consider a quasilinear viscoelastic wave equation with acoustic boundary conditions. Under some appropriate assumption on the relaxation function g, the function Φ, p > max { ρ + 2 , m , q , 2 } $p > \max \{ \rho +2, m, q,2\}$ , and the initial data, we prove a global nonexistence of solutions for a quasilinear viscoelastic wave equation with positive initial...