In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: $$\begin{aligned} \textstyle\begin{cases} (-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{\mu }{u^{\gamma }}+f & \text{in } \Omega, \\ u>0 & \text{in } \Omega, \\ u=0 & \text{in } (\mathbb{R}^{N} \setminus \Omega

The major goal of this work is investigating sufficient conditions for the existence and uniqueness of solutions for implicit impulsive coupled system of φ-Hilfer fractional differential equations (FDEs) with instantaneous impulses and terminal conditions. First, we derive equivalent fractional integral equations of the proposed system. Next, by employing some standard fixed point theorems such as

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain

The Editors-in-Chief have retracted this article because it shows significant overlap with an article by different authors that was simultaneously under consideration with another journal [1]. The article also shows evidence of authorship manipulation and peer review manipulation. The authors have not responded to correspondence regarding this retraction. 1. Luan, K., Vieira, J.: Poisson-type inequalities

In this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

In this article, we deal with a strongly damped von Karman equation with variable exponent source and memory effects. We investigate blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time.

This note obtains a new regularity criterion for the three-dimensional magneto-micropolar fluid flows in terms of one velocity component and the gradient field of the magnetic field. The authors prove that the weak solution $(u,\omega,b)$ to the magneto-micropolar fluid flows can be extended beyond time $t=T$ , provided if $u_{3}\in L^{\beta }(0,T;L^{\alpha }(R^{3}))$ with $\frac{2}{\beta }+\frac{3}{\alpha

This article is devoted to a study of the blow-up result for a system of coupled viscoelastic wave equations. By establishing a new auxiliary function and using the reduction to absurdity method, we obtain some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. This work generalizes and improves earlier results in the literature

Three new and applicable approaches based on quasi-linearization technique, wavelet-homotopy analysis method, spectral methods, and converting two-point boundary value problem to Fredholm–Urysohn integral equation are proposed for solving a special case of strongly nonlinear two-point boundary value problems, namely Troesch problem. A quasi-linearization technique is utilized to reduce the nonlinear

Under the acoustic boundary conditions, the initial boundary value problem of a wave equation with multiple nonlinear source terms is considered. This paper gives the energy functional of regular solutions for the wave equation and proves the decreasing property of the energy functional. Firstly, the existence of a global solution for the wave equation is proved by the Faedo–Galerkin method. Then,

This paper is devoted to studying the following nonlinear fractional problem: 0.1 $$ \textstyle\begin{cases} (-\Delta )^{s}u+u=K( \vert x \vert )u^{p},\quad u>0, x\in {\mathbb{R}}^{N}, \\ u(x)\in H^{s}({\mathbb{R}}^{N}), \end{cases} $$ where $N\geq 3$ , $0< s<1$ , $1< p<\frac{N+2s}{N-2s}$ , $K(|x|)$ is a positive radical function. We constructed infinitely many non-radial solutions of the new type

In this paper, we consider the problem of free convection in a square cavity filled with a porous medium with convective boundary condition on the left wall of the cavity. We first transform the governing equations transformed in terms of dimensionless variables and then solve them numerically using a cubic spline colocation method. We discuss the effects of two very important parameters, the Biot

In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor $\{A_{\varepsilon }(t)\}_{t\in \mathbb{R}}$ of Eq. (1.1) with $\varepsilon \in [0,1]$ satisfies $\lim_{\varepsilon \to \varepsilon _{0}}\sup_{t\in [a,b]} \operatorname{dist}_{H_{0}^{1}\times L

In this paper, we study the following nonlocal problem: $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ with $N\ge 3$ , $a,b>0$ , $1< q<2$ and $\lambda >0$ is a parameter. By virtue of the variational method

The Editors-in-Chief have retracted this article because it shows significant overlap with an article that was simultaneously under consideration [1]. Additionally, the article shows evidence of peer review manipulation. The author has not responded to correspondence regarding this retraction. 1. Xue, G.: Rarefied sets at infinity associated with the Schrödinger operator. J Inequal Appl 2014, 247 (2014)

We show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ and $M>0$ such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$

We investigate the nonlinear Rayleigh–Taylor (RT) instability of a nonhomogeneous incompressible nematic liquid crystal in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution. Thus we construct solutions of the linearized problem that grow in time in the Sobolev space $H^{4}$ , then we show that the RT equilibrium state is linearly

The Editors-in-Chief have retracted this article because it shows significant overlap with a previously published article by Guo and Ye [1] and an article by Meng that was simultaneously under consideration [2]. The author has not responded to correspondence regarding this retraction. 1. Guo, Q., Ye, P.: Error analysis for lq-coefficient regularized moving least-square regression. J Inequal Appl 2018

The topological sensitivity method is an optimization technique used in different inverse problem solutions. In this work, we adapt this method to the identification of plasma domain in a Tokamak. An asymptotic expansion of a considered shape function is established and used to solve this inverse problem. Finally, a numerical algorithm is developed and tested in different configurations.

In this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

This paper presents a numerical algorithm for solving high-order BVPs. We introduce the construction method of multiscale orthonormal basis in $W^{m}_{2}[0,1]$ . Based on the orthonormal basis, the numerical solution of the boundary value problem is obtained by finding the ε-approximate solution. In addition, the convergence order, stability, and time complexity of the method are discussed theoretically

We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

The structural stability for the Forchheimer fluid interfacing with a Darcy fluid in a bounded region in $\mathbb{R}^{3}$ was studied. We assumed that the nonlinear fluid was governed by the Forchheimer equations in $\Omega _{1}$ , while in $\Omega _{2}$ , we supposed that the flow satisfies the Darcy equations. With the aid of some useful a priori bounds, we were able to demonstrate the continuous

In this paper, we study the following quasilinear Schrödinger equation: $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ where $N\geq 3$ , $1< p< N$ , $-\infty <\alpha <\frac{N-p}{p}$ , $\alpha \leq e\leq \alpha +1$ , $d=1+\alpha -e$ , $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$

In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show

A diffusive phytoplankton–zooplankton model with nonlinear harvesting is considered in this paper. Firstly, using the harvesting as the parameter, we get the existence and stability of the positive steady state, and also investigate the existence of spatially homogeneous and inhomogeneous periodic solutions. Then, by applying the normal form theory and center manifold theorem, we give the stability

In this paper, we consider the linear stability of blowup solution for incompressible Keller–Segel–Navier–Stokes system in whole space $\mathbb{R}^{3}$ . More precisely, we show that, if the initial data of the three dimensional Keller–Segel–Navier–Stokes system is close to the smooth initial function $(0,0,\textbf{u}_{s}(0,x) )^{T}$ , then there exists a blowup solution of the three dimensional linear

The aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$ . We first derive a critical inequality in $H^{n}$ . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear

By employing critical point theory, we investigate the existence of solutions to a boundary value problem for a p-Laplacian partial difference equation depending on a real parameter. To be specific, we give precise estimates of the parameter to guarantee that the considered problem possesses at least three solutions. Furthermore, based on a strong maximum principle, we show that two of the obtained

In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.

Drift rate is a very important parameter in the evolution of stock price, which has significant impact on the corresponding option pricing. This paper deals with an inverse problem of recovering the drift function by current market prices of options. Different from the usual inverse volatility problem, our mathematical model does not tend to zero at infinity, which may bring great trouble to theoretical

In this paper, we establish the Fujita type theorem for a homogeneous Neumann outer problem of the coupled quasilinear convection–diffusion equations and formulate the critical Fujita exponent. Besides, the influence of diffusion term, reaction term, and convection term on the global existence and the blow-up property of the problem is revealed. Finally, we discuss the large time behavior of the solution

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results

In the paper, we study a boundary value problem for a class of ψ-Hilfer fractional-order Langevin equations with multi-point integral boundary conditions. Existence and uniqueness results are established by using well-known fixed point theorems. Examples illustrating the main results are also included.

In this paper we investigate the existence of traveling wave for a one-dimensional reaction diffusion system. We show that this system has a unique translation traveling wave solution.

By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.

The aim of this paper is to investigate the optimal harvesting strategies of a stochastic competitive Lotka–Volterra model with S-type distributed time delays and Lévy jumps by using ergodic method. Firstly, the sufficient conditions for extinction and stable in the time average of each species are established under some suitable assumptions. Secondly, under a technical assumption, the stability in

In this paper, we consider the following fractional Kirchhoff problem with strong singularity: $$ \textstyle\begin{cases} (1+ b\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}} \frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3+2s}}\,\mathrm{d}x \,\mathrm{d}y )(-\Delta )^{s} u+V(x)u = f(x)u^{-\gamma }, & x \in \mathbb{R}^{3}, \\ u>0,& x\in \mathbb{R}^{3}, \end{cases} $$ where $(-\Delta )^{s}$ is the

In this paper, we study the parabolic Monge–Ampère equations $-u_{t}\det (D^{2}u)=g$ outside a bowl-shaped domain with g being the perturbation of $g_{0}(|x|)$ at infinity. Under the weaker conditions compared with the problem outside a cylinder, we obtain the existence and uniqueness of viscosity solutions with asymptotic behavior for the first initial-boundary value problem by using the Perron method

We study various types of uniform Calderón–Zygmund estimates for weak solutions to elliptic equations in periodic homogenization. A global regularity is obtained with respect to the nonhomogeneous term from weighted Lebesgue spaces, Orlicz spaces, and weighted Orlicz spaces, which are generalized Lebesgue spaces, provided that the coefficients have small BMO seminorms and the domains are δ-Reifenberg

A priori bounds were derived for the flow in a bounded domain for the viscous-porous interfacing fluids. We assumed that the viscous fluid was slow in $\Omega _{1}$ , which was governed by the Boussinesq equations. For a porous medium in $\Omega _{2}$ , we supposed that the flow satisfied the Darcy equations. With the aid of these a priori bounds we were able to demonstrate the result of the continuous

This paper is dedicated to studying the following Kirchhoff-type problem: $$ \textstyle\begin{cases} -m ( \Vert \nabla u \Vert ^{2}_{L^{2}(\mathbb{R} ^{N})} )\Delta u+V(x)u=f(u), & x\in \mathbb{R} ^{N}; \\ u\in H^{1}(\mathbb{R} ^{N}), \end{cases} $$ where $N=1,2$ , $m:[0,\infty )\rightarrow (0,\infty )$ is a continuous function, $V:\mathbb{R} ^{N}\rightarrow \mathbb{R} $ is differentiable, and $f\in

In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the $(\alpha ,\beta )$ -resolvent operator, we concern with the term $u'(\cdot )$ and finding a control v such that the mild solution satisfies $u(b)=u_{b}$ and $u'(b)=u'_{b}$

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1186/s13661-021-01502-z

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the $L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})$ -norm of a suitable low frequency part of $(u,\nabla d)$ is bounded by a scaling invariant constant and the initial data $(u_{0},\nabla d_{0})$ . Our result refines the corresponding one in (Liu and Zhao in J. Math.

In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.

We consider the inertial Qian–Sheng’s Q-tensor dynamical model for the nematic liquid crystal flow, which can be viewed as a system coupling the hyperbolic-type equations for the Q-tensor parameter with the incompressible Navier–Stokes equations for the fluid velocity. We prove the existence and uniqueness of local in time strong solutions to the system with the initial data near uniaxial equilibrium

In this paper, we consider a class of fractional boundary value problems with the derivative term and nonlinear operator term. By establishing new mixed monotone fixed point theorems, we prove these problems to have a unique solution, and we construct the corresponding iterative sequences to approximate the unique solution.

In this article we extend the known theory of solution regions to encompass nonlinear boundary conditions. We both provide results for new boundary conditions and recover some known results for the linear case.

We study the quasilinear elliptic problem which is resonant at zero. By using Morse theory, we obtain five nontrivial solutions for the equation with coercive nonlinearities.

Consider a double degenerate parabolic equation arising from the electrorheological fluids theory and many other diffusion problems. Let $v_{\varepsilon }$ be the viscous solution of the equation. By showing that $|\nabla v_{\varepsilon }|\in L^{\infty }(0,T; L_{\mathrm{loc}}^{p(x)}(\Omega ))$ and $\nabla v_{\varepsilon }\rightarrow \nabla v$ almost everywhere, the existence of weak solutions is proved

A type of non-Newtonian filtration equations with variable delay is considered. Using a new approach which was established by Ge and Ren in (Nonlinear Anal. 58:477–488, 2004), we obtain the existence of periodic wave solutions for the non-Newtonian filtration equations. The methods of the present paper are markedly different from the existing ones.

We study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$ where $\lambda >0$ is a positive parameter

In this paper, we prove the existence of random $\mathcal{D}$ -attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the random $\mathcal{D}$ -attractor.

We study the coupled Choquard type system with lower critical exponents $$ \textstyle\begin{cases} -\Delta u+\lambda _{1}(x)u=\mu _{1}(I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u+\beta (I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u,\quad x\in {\mathbb{R}}^{N}, \\ -\Delta v+\lambda _{2}(x)v=\mu _{2}(I_{\alpha }*

A stochastic four species food-chain model is proposed in this paper. Here, artificial harvest in each species and the effect of time delay for interaction between species are considered, which makes the model more applicable in real situations. Specifically, we address the stochastic global dynamics behavior, including the existence of global positive solutions, stochastic ultimate boundedness, extinction

In this paper, we study the global $C^{1, 1}$ regularity for viscosity solution of the degenerate Monge–Ampère type equation $\det [D^{2}u-A(x, Du)]=B(x, u, Du)$ with the Neumann boundary value condition $D_{\nu }u=\varphi (x)$ , where the matrix A is under the regular condition and some structure conditions, and the right-hand term B is nonnegative.

This paper deals with the asymptotic behavior of solutions to the initial-boundary value problem of the following fractional p-Kirchhoff equation: $$ u_{t}+M\bigl([u]_{s,p}^{p}\bigr) (-\Delta )_{p}^{s}u+f(x,u)=g(x)\quad \text{in } \Omega \times (0, \infty ), $$ where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with Lipschitz boundary, $N>ps$ , $0< s<1

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1186/s13661-021-01488-8