In this paper, we study the following Choquard type quasilinear Schrödinger equation: $$ -\Delta u+u-\Delta \bigl(u^{2}\bigr)u=\bigl(I_{\alpha }*G(u) \bigr)g(u),\quad x\in {\mathbb{R}}^{N}, $$ where $N\geq 3$ , $0<\alpha

We are concerned with the following elliptic equations with variable exponents: $$ M \bigl([u]_{s,p(\cdot,\cdot)} \bigr)\mathcal{L}u(x) +\mathcal {V}(x) \vert u \vert ^{p(x)-2}u =\lambda\rho(x) \vert u \vert ^{r(x)-2}u + h(x,u) \quad \text{in } \mathbb {R}^{N}, $$ where $[u]_{s,p(\cdot,\cdot)}:=\int_{\mathbb {R}^{N}}\int_{\mathbb {R}^{N}} \frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)|x-y|^{N+sp(x,y)}} \,dx \

We introduce the investigation of approximate controllability for a new class of nonlocal and noninstantaneous impulsive Hilfer fractional neutral stochastic integrodifferential equations with fractional Brownian motion. An appropriate set of sufficient conditions is derived for the considered system to be approximately controllable. For the main results, we use fractional calculus, stochastic analysis

In this paper, we consider the nonlocal controllability of $\alpha\in (1,2)$ -order fractional evolution systems of Sobolev type in abstract spaces. By utilizing fixed point theorems and the theory of resolvent operators we establish some sufficient conditions for the nonlocal controllability of Sobolev-type fractional evolution systems.

This paper deals with the existence of multiple solutions for the following Kirchhoff type equations involving p-biharmonic operator: $$\begin{aligned}& -M \biggl( \int_{\varOmega} \bigl( \vert \Delta_{p}u \vert ^{2}+ \vert u \vert ^{p} \bigr)\,dx \biggr) \bigl( \Delta _{p}^{2}u- \vert u \vert ^{p-2}u \bigr) =\lambda f(x) \vert u \vert ^{q-2}u+g(x) \vert u \vert ^{m-2}u,\quad x\in\varOmega, \end{aligned}$$

In this paper, without requiring the complete continuity of integral operators and the existence of upper–lower solutions, by means of the sum-type mixed monotone operator fixed point theorem based on the cone $P_{h}$, we investigate a kind of p-Laplacian differential equation Riemann–Stieltjes integral boundary value problem involving a tempered fractional derivative. Not only the existence and uniqueness

In this paper, we prove that the existence of global attractors for a Kirchhoff wave equation with nonlinear damping and memory.

In this paper, a priori estimate for a linear third pseudoparabolic operator with bound is established, and applying the above result, the existence and uniqueness theorem of solutions for a class of nonlinear pseudoparabolic equations is obtained with the help of the homeomorphism method and the initial value method. Furthermore, an existence and uniqueness theorem of the semilinear equation is obtained

The existence of a solution for a second-order p-Laplacian boundary value problem at resonance with two dimensional kernel will be considered in this paper. A semi-projector, the Ge and Ren extension of Mawhin’s coincidence degree theory, and algebraic processes will be used to establish existence results, while an example will be given to validate our result.

For arbitrary two-point boundary condition, which makes the corresponding linear uniform problem well-posed, we obtain an existence and uniqueness result for the boundary value problem of finite beam deflection resting on arbitrary nonlinear non-uniform elastic foundation. The difference between the desired solution and the corresponding linear uniform one in $L^{\infty}$ sense is bounded explicitly

In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity: $$ (-\Delta )_{A}^{s}u+V(x)u=f\bigl(x, \vert u \vert ^{2}\bigr)u+\lambda \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, $$ where $(-\Delta )_{A}^{s}$ is the fractional magnetic operator with $0< s<1$, $N>2s$, $\lambda >0$, $2_{s}^{*}=\frac{2N}{N-2s}$, $p\geq

This paper deals with the following Kirchhoff–Schrödinger–Poisson system: $$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)f(u)&\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}, \end{cases} $$ where a, b are positive constants, $K(x)$, $V(x)$ are positive continuous functions vanishing at infinity, and $f(u)$ is

On the interval $[0,1]$ we consider the nth order linear differential equation, the coefficient of the highest derivative of which is equivalent to the power function $t^{\mu }$ when $t\rightarrow 0$. The main aim of the paper is to pose “generalized” Cauchy conditions for the given equation at the point of singularity $t=0$, which would be correct for any $\mu >0$.

The aim of this paper is establishing the existence of a nontrivial solution for the following quasilinear Schrödinger–Poisson system: $$ \left \{ \textstyle\begin{array}{l} -\Delta u+V(x)u-u\Delta(u^{2})+K(x)\phi(x)u=g(x, u),\quad x\in\mathbb {R}^{3}, \\ -\Delta\phi=K(x)u^{2}, \quad x\in\mathbb{R}^{3},\\ u\in H^{1}(\mathbb{R}^{3}),\qquad u>0, \end{array}\displaystyle \right . $$ where V, K, g are

The present paper deals with a class of nonlocal problems. Under some suitable assumptions on the decay rate of the coefficients, we derive the existence of infinitely many positive solutions to the problem by applying reduction method. Comparing to the previous work, we encounter some new challenges because of competing potentials. By doing some delicate estimates for the competing potentials, we

This paper deals with the global dynamics for a SLIS epidemic model with infection age. In our model, we also consider the time delay in the progression from the latent individuals to becoming infectious individuals. We verify the well posedness of the model by changing it into an abstract nondensely defined Cauchy problem and find conditions for the existence of disease free equilibrium and endemic

We consider the existence of multiple solutions of the following singular nonlocal elliptic problem: $$\begin{aligned} \textstyle\begin{cases} -M(\int _{\mathbb{R} ^{N}}{ \vert x \vert ^{-ap} \vert \nabla u \vert ^{p}})\operatorname{div}( \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\nabla u)= h(x) \vert u \vert ^{r-2}u+H(x) \vert u \vert ^{q-2}u, \\ u(x)\rightarrow 0 \quad \text{as } \vert x \vert

For the nonlinear predator–prey system (PPS), although a variety of numerical methods have been proposed, such as the difference method, the finite element method, and so on, but the efficient numerical method has always been the direction that scholars strive to pursue. Based on this question, a sinc function interpolation method is proposed for a class of PPS. Numerical simulations of a class of

An important stage in an analysis of a multibody system (MBS) with elastic elements by the finite element method is the assembly of the equations of motion for the whole system. This assembly, which seems like an empirical process as it is applied and described, is in fact the result of applying variational formulations to the whole considered system, putting together all the finite elements used in

In this paper, we consider the primitive three-dimensional viscous equations for large-scale atmosphere dynamics with topography effects and water vapor phase transition process. This modified climate model is commonly used in weather and climate predictions, and few theoretical analyses have been performed on them. The existence and uniqueness of a global strong solution to this climate model is established

We investigate a nonlinear generalized Fornberg–Whitham equation. The key element is that we derive an $L^{2}(\mathbb{R})$ conservation law for solutions of the equation. We establish several estimates by utilizing the $L^{2}(\mathbb{R})$ conservation law. These estimates lead to the proof of the existence and uniqueness of entropy weak solution of the equation in the space $L^{1}(\mathbb{R})\cap

This paper is concerned with the inverse scattering of time-harmonic acoustic plane waves by a multi-layered fluid–solid medium in the three dimensional space. We establish the global uniqueness in identifying the embedded penetrable solid obstacle, the surrounding fluid medium and its wave number from the acoustic far-field pattern for all incident plane waves at a fixed frequency. The proof depends

The Editors-in-Chief have retracted this article [1] because it significantly overlaps with a number of previously published articles from different authors [2–4]. The author has not responded to any correspondence regarding this retraction.

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg

This research work is dedicated to investigating a class of impulsive fractional order differential equations under the Robin boundary conditions via the application of topological degree theory (TDT). We establish some adequate results for the existence of at most one solution for the consider problem. Further, the whole analysis is illustrated by providing a pertinent example. We keep in mind that

In this paper, we study a nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions. By using some new techniques, we introduce a formula of solutions for above problem, which can be regarded as a novelty item. Moreover, under the weak assumptions and using Leray–Schauder degree theory, we obtain the existence result of solutions for above problem

The global solution of the $n \geq3$ Landau–Lifshitz–Gilbert equation on $\mathbb{S}^{2}$ is studied under the cylindrical symmetric coordinates. An equivalent complex-valued equation in cylindrical symmetric coordinates is obtained by the Hasimoto transformation. A renormalization for the Laplacian is used to transform this equivalent system to a Ginzberg–Landau type system in which the Strichartz

In this paper, based on the notation of time-dependent attractors introduced by Conti, Pata and Temam in (J. Differ. Equ. 255:1254–1277, 2013), we prove the existence of time-dependent global attractors in $\mathcal{H}_{t}$ for a class of nonclassical reaction–diffusion equations with the forcing term $g(x)\in H^{-1}(\varOmega )$ and the nonlinearity f satisfying the polynomial growth of arbitrary

This paper deals with the implementation of the spectral discretization of the vorticity–velocity–pressure formulation of the nonstationary Stokes problem. We use the implicit Euler scheme for the discretization in time. We propose a global method for the resolution of the linear system resulting from the discrete problem. This method is implemented, and some numerical results are presented which confirm

In this paper, we develop optimal Phragmén–Lindelöf methods, based on the use of maximum modulus of optimal value of a parameter in a Schrödinger functional, by applying the Phragmén–Lindelöf theorem for a second-order boundary value problems with respect to the Schrödinger operator. Using it, it is possible to find the existence of ground state solutions of the generalized Schrödinger equation with

We discuss the existence and uniqueness of solution for the second boundary value problem of potential theory often referred to as the Neumann problem, on a gauge ball for the canonical sub-Laplacian in H-type groups. In this way we extend the classical results of the problem as well as its generalization to the Heisenberg group.

In this paper, a fast multiscale Galerkin algorithm is developed for solving the boundary value problem of the fractional Bagley–Torvik equation. For this purpose, we employ multiscale orthogonal functions having vanishing moments as the basis of the trial space, and we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which leads to a fast algorithm. We

The paper studies a system of nonlinear viscoelastic Kirchhoff system with a time varying delay and general coupling terms. We prove the global existence of solutions in a bounded domain using the energy and Faedo–Galerkin methods with respect to the condition on the parameters in the coupling terms together with the weight condition as regards the delay terms in the feedback and the delay speed. Furthermore

The main purpose of this paper is to investigate the existence of a positive periodic solution for a prescribed mean curvature generalized Liénard equation with a singularity (weak and strong singularities of attractive type, or weak and strong singularities of repulsive type). Our proof is based on an extension of Mawhin’s continuation theorem.

The existence of at least one positive radial solution of the Neumann problem $$ -\Delta _{\mathbb{H}^{n}} u+R(\xi ) u=a \bigl( \vert \xi \vert _{\mathbb{H}^{n}} \bigr) \vert u \vert ^{p-2} u - b\bigl( \vert \xi \vert _{\mathbb{H}^{n}}\bigr) \vert u \vert ^{q-2}u, $$ is proved on the Heisenberg group $\mathbb{H}^{n}$, via the variational principle, where $a(|\xi |_{\mathbb{H}^{n}})$, $b(|\xi |_{\mathbb{H}^{n}})$

We study the existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a new cone to establish the existence of solutions by means of fixed point index theory.

The Editors-in-Chief have retracted this article [1] because it significantly overlaps with an article from other authors that was simultaneously under consideration at another journal [2]. Additionally, the article also shows evidence of authorship and peer review manipulation. The authors have not responded to any correspondence regarding this retraction.

In this paper, we consider the initial boundary value problem for generalized Zakharov equations. Firstly, we prove the existence and uniqueness of the global smooth solution to the problem by a priori integral estimates, the Galerkin method, and compactness theory. Furthermore, we discuss the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero

In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies $$ h^{\prime}(t)\le-\xi(t) H\bigl(h(t)\bigr),\quad t\geq0, $$ where the

This paper studies the existence of affine-periodic solutions which have the form of $x(t+T)=Qx(t)$ with some nonsingular matrix Q. Depending on the structure of Q, they can be periodic, anti-periodic, quasi-periodic or even unbounded. Krasnosel’skii–Perov type existence theorem, asymptotic and homotopy equivalence approaches are given.

This paper is devoted to the existence and non-existence of positive solutions to the following negative power nonlinear integral equation related to the sharp reversed Hardy–Littlewood–Sobolev inequality: $$ f^{q-1}(x)= \int _{\varOmega }\frac{K(x)f(y)K(y)}{ \vert x-y \vert ^{n-\alpha }}\,dy+ \lambda \int _{\varOmega }\frac{G(x)f(y)G(y)}{ \vert x-y \vert ^{n-\alpha -\beta }}\,dy, \quad f\geq 0, x\in

This paper aims to investigate the class of quasilinear Schrödinger equations 0.1 $$\begin{aligned} \begin{aligned}[b] &-\Delta u-\bigl[\Delta \bigl(1+u^{2}\bigr)^{\frac{\gamma }{2}}\bigr] \frac{\gamma u}{2(1+u^{2})^{\frac{2-\gamma }{2}}}\\ &\quad =\alpha h\bigl( \vert x \vert \bigr) \vert u \vert ^{p-1}u+ \beta H\bigl( \vert x \vert \bigr) \vert u \vert ^{q-1}u, \quad x\in \mathbb{R}^{N}, \end{aligned}

Hepatitis B virus (HBV) is a life-threatening virus that causes very serious liver-related diseases from the family of Hepadnaviridae having very rare qualities resembling retroviruses. In this paper, we analyze the effect of antiviral therapy through mathematical modeling by using Liao’s homotopy analysis method (LHAM) that defines the connection between the target liver cells and the HBV. We also

The non-autonomous order-2γ parabolic partial differential equation for an epitaxial thin film growth model with dimension $d=3$ is investigated by the method of uniform estimates. The existence of a pullback attractor for the 3D model is proved for $\gamma >3$.

The Editors-in-Chief have retracted this article [1] because it significantly overlaps with a previously published article by Pang et al. [2]. The article also shows evidence of authorship manipulation. The identity of the corresponding author could not be verified; the University of Ioannina have confirmed that Mohamed Vetro has not been affiliated with their institution. The authors have not responded

In this paper, we focus on the class of almost-periodically forced higher-dimensional beam equations $$ u_{tt}+(-\Delta +\mu )^{2}u+\psi (\omega t)u=0,\quad \mu >0, t \in \mathbb{R}, x\in \mathbb{R}^{d}, $$ subject to periodic boundary conditions, where $\psi (\omega t)$ is real analytic and almost-periodic in t. We show the existence of almost-periodic solutions for this equation under some suitable

In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain $(\varOmega\in \mathbb{R}^{n}, n\geq 3)$ with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.

In this article, we study a modified maximum principle approach under condition on the weight of the delay term in the feedback and the weight of the term without delay. On that basis, we prove the existence of global solutions for a quasilinear Schrödinger equation in an unbounded domain with a general nonlinear nonlinear optimal control condition in the weakly nonlinear internal feedback. The equation

The stochastic coupled Kuramoto–Sivashinsky and Ginzburg–Landau equations (KS-GL) perturbed by additive noises is investigated in this paper. By making careful analysis, we first consider the existence and uniqueness of the solution with initial-boundary condition, and then we establish a random attractor for the stochastic KS–GL equations in $X=L^{2} \times H^{-1}$.

We consider a class of generalized quasilinear Schrödinger equations $$ -\operatorname{div}\bigl(l^{2}(u)\nabla u\bigr)+l(u)l'(u) \vert \nabla u \vert ^{2}+V(x)u= f(u),\quad x\in \mathbb{R}^{N}, $$ where $l(t): \mathbb{R}\to\mathbb{R}^{+}$ is a nondecreasing function with respect to $|t|$, the potential function V is allowed to be sign-changing so that the Schrödinger operator $-\Delta+V$ possesses

In this paper, we consider the egg-sperm chemotaxis model of coral with the incompressible fluid equations in the whole space. The existence of global mild solutions in scaling invariant spaces is proved with sufficient small initial data. Here the main tool we use is the implicit function theorem. Furthermore, we obtain the asymptotic stability of solutions when the time goes to infinity. Since the

This paper considers a two-unit complex system, in which one of the components has priority with preemptive repeat repair disciplines, described by partial differential equations with integral boundary conditions. First, we prove that the system has a unique nonnegative time-dependent solution by using the strong continuous semigroup theory of linear operators. Then, we obtain the exponential convergence

In this paper, the existence of a response solution with the Liouvillean frequency vector to the quasi-periodically forced complex Ginzburg–Landau equation, whose linearized system is elliptic–hyperbolic, is obtained. The proof is based on constructing a modified KAM theorem for an infinite-dimensional dissipative system with Liouvillean forcing frequency.

A degenerate parabolic equation of the form $$\bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+\nabla\vec{g}\cdot\nabla\vec{\gamma}(v) $$ is considered, where $\vec{g}=\{g^{i}(x,t)\}$, $\vec{\gamma}(v)=\{ \gamma_{i}(v)\}$. If the diffusion coefficient $b(x,t)\geq0$ is degenerate on the boundary, by adding some restrictions on

In the paper, we investigate global and blow-up solutions for a class of nonlinear reaction diffusion equations with Robin boundary conditions. By using auxiliary functions and a first-order differential inequality technique, we establish conditions on the data to prove the existence of global solution. Moreover, based on maximum principles, we obtain a criterion that guarantees the occurrence of the

In this paper, a class of boundary value problems for fractional differential equations with a parameter is studied via the variational methods. Firstly, we present a result that the boundary value problems have at least one weak solution under the quadratic condition and the superquadratic condition, respectively. Additionally, we obtain the existence of at least one nontrivial solution by using the

In this paper, a class of quasilinear Schrödinger equations with discontinuous nonlinearity is considered. After changing variables, by using nonsmooth critical point theory, we obtain the existence and concentration of positive solutions for this problem under suitable conditions. Our results cover and extend some results for these differentiable quasilinear Schrödinger problems.

The aim of this article is to consider the semi-linear fractional system with Sobolev exponents $q = \frac{{n + \alpha}}{{n - \beta}}$ and $p = \frac{{n + \beta}}{{n - \alpha}}$ ($\alpha \ne\beta$): $$\left \{ \textstyle\begin{array}{l} {( - \Delta)^{\alpha/2}}u(x) = k(x){v^{q}}(x) + f(v(x)),\\ {( - \Delta)^{\beta/2}}v(x) = j(x){u^{p}}(x) + g(u(x)), \end{array}\displaystyle \right . $$ where $0 < \alpha

We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued

In this paper, we investigate the existence of solutions for the fractional p-Laplace equation $$ (-\Delta)_{p}^{s}u+V(x) \vert u \vert ^{p-2}u=h_{1}(x) \vert u \vert ^{q-2}u+h_{2}(x) \vert u \vert ^{r-2}u \quad \mbox{in } \mathbb{R}^{N}, $$ where $N>sp$, $0< s<10$ and $h_{1}(x)$, $h_{2}(x)$ are allowed to change sign in $\mathbb {R}^{N}$. By using variant fountain theorem, we prove that the above