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This paper serves as a corrigendum to the paper [Electron. J. Qual. Theor. Differ. Equ. 2018, No. 84, 1–14]. We present here a corrected version of Theorem 3.1 and Theorem 5.1.

This addendum concerns the paper of the above title found in EJQTDE No. 57 (2011). The example in Section 4 was not correct. The following example is a correction given by the authors. We regret any inconvenience which this may have caused any reader.

In this paper, we study the following Schrödinger equations with potentials and general nonlinearities \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta f(u), \\ \int |u|^2dx=\Theta, \end{cases} \end{equation*} both on $\mathbb{R}^N$ as well as on domains $\Omega_r$ where $\Omega_r \subset \mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponent satisfies $2+\frac{4}{N}\leq q\leq2^*=\frac{2 N}{N-2}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $L^2$-subcritical or $L^2$-critical growth. This paper generalizes the conclusion of Bartsch et al. in [4]. Moreover, we consider the Sobolev critical case and $L^2$-critical case of the above problem.

In this paper, we investigate the bifurcation results of the fractional Kirchhoff–Schrödinger–Poisson system \begin{equation*} \begin{cases} M([u]_s^2)(-\Delta)^s u+V(x)u+\phi(x) u=\lambda g(x)|u|^{p-1}u+|u|^{2_s^*-2}u~~&{\rm in}~\mathbb{R}^3, \\ (-\Delta)^t \phi(x)=u^2~~&{\rm in}~\mathbb{R}^3, \end{cases} \end{equation*} where $s,t\in(0,1)$ with $2t+4s>3$ and the potential function $V$ is a continuous function. We show that the existence of components of (weak) solutions of the above equation bifurcates out from the first eigenvalue $\lambda_1$ of the problem $$(-\Delta)^s u+V(x)u=\lambda g(x)u\quad\mbox{in }\mathbb R^3.$$ The main feature of this paper is the inclusion of a potentially degenerate Kirchhoff model, combined with the critical nonlinearity.

In this paper, we introduce logistic equations with Stieltjes derivatives and provide explicit solution formulas. As an application, we present a population model which involves intraspecific competition, periods of hibernation, as well as seasonal reproductive cycles. We also deal with various forms of Stieltjes integral equations, and find the corresponding logistic equations. We show that our work extends earlier results for dynamic equations on time scales, which served as an inspiration for this paper.

This paper stands for the almost sure practical stability of nonlinear stochastic differential delay equations (SDDEs) with a general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functionals. Furthermore, we provide some numerical examples to validate the effectiveness of the abstract results of this paper.

Little work seems to be known about stabilization results of highly nonlinear stochastic time-varying delay systems (STVDSs) with Markovian switching and Poisson jump. This paper is concerned with the stabilization problem for a class of STVDSs with Markovian switching and Poisson jump. The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. The aim of this paper is to design a delay feedback controller to make an unstable highly nonlinear STVDSs with Markovian switching and Poisson jump $H_\infty$-stable and asymptotically stable. Besides, an illustrative example is provided to support the theoretical results.

In this paper, we utilize the Littlewood–Paley decomposition theory to establish two regularity criteria for the 3D magneto-micropolar equations in Vishik spaces, specifically focusing on the gradient of the velocity field.

In this paper, ground-state solutions to a Hartree–Fock type system with a critical growth are studied. Firstly, instead of establishing the local Palais–Smale (P.-S.) condition and estimating the mountain-pass critical level, a perturbation method is used to recover compactness obtain the existence of ground-state solutions. To achieve this, an important step is to get the right continuity of the mountain-pass level on the coefficient in front of perturbing terms. Subsequently, depending on the internal parameters of coupled nonlinearities, whether the ground state is semi-trivial or vectorial is proved.

Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi: \mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[% \begin{cases} -\phi\left( u^{\prime}\right) ^{\prime}=\lambda m(x)f(u)+\mu n(x)g(u) & \text{ in }\Omega,\\ u=0 & \text{ on }\partial\Omega, \end{cases} \] where $f,g:[0,\infty)\rightarrow\lbrack0,\infty)$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $\phi$, respectively. Our assumptions on $\phi$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo–Krasnoselskiĭ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems.

In this paper, a diffusive predator-prey system considering prey-taxis term with memory and maturation delays under Neumann boundary conditions is investigated. Firstly, the existence and stability of equilibria, especially the existence, uniqueness and stability of the positive equilibrium, are studied. Secondly, we prove that: $\rm (i)$ there is no spatially homogeneous steady state bifurcation as the eigenvalue of the negative Laplace operator is zero; $\rm (ii)$ as this system is only with memory delay $\tau_{1}$, the the spatially nonhomogeneous Hopf bifurcation appears; $\rm (iii)$ when the model is only with maturation delay $\tau_{2}$, the system has spatially homogeneous and nonhomogeneous periodic solutions; $\rm (iv)$ for the case of two delays, the system has rich dynamics, for example, stability switches, whose curves have four forms. Finally, some numerical simulations are produced to verify and support the theoretical results.

In this paper, we are devoted to establishing that the existence of positive solutions for a class of generalized quasilinear elliptic equations in $\mathbb{R}^{N}$ with Sobolev critical growth, which have appeared from plasma physics, as well as high-power ultrashort laser in matter. To begin, by changing the variable, quasilinear equations are transformed into semilinear equations. The positive solutions to semilinear equations are then presented using the Mountain Pass Theorem for locally Lipschitz functionals and the Concentration-Compactness Principle. Finally, an inverse translation reveals the presence of positive solutions to the original quasilinear equations.

This paper studies the global phase portraits of uniform isochronous quintic centers at the origin with time reversibility such that their nonlinear part is not homogeneous. By using Poincaré compactification, we obtain all possible phase portraits of this quintic polynomial differential system.

The aim of this paper is to estimate an interval of the existence of positive solutions for a boundary value problem for the system of three nonlinear second-order ordinary differential equations. Krasnosel'skiĭ–Precup fixed point theorem is used to determine this interval theoretically. For Dirichlet boundary conditions theoretical result is compared with result obtained numerically.

This paper considers the existence of multiple normalized solutions of the following $(2,q)$-Laplacian equation: \begin{equation*} \begin{cases} -\Delta u-\Delta_q u=\lambda u+h(\epsilon x)f(u), &\mathrm{in}\ \mathbb{R}^{N},\\ \int_{\mathbb{R}^{N}}|u|^2\mathrm{d}x=a^2, \end{cases} \end{equation*} where $20, a>0$ and $\lambda \in \mathbb{R}$ is a Lagrange multiplier which is unknown, $h$ is a continuous positive function and $f$ is also continuous satisfying $L^2$-subcritical growth. When $\epsilon$ is small enough, we show that the number of normalized solutions is at least the number of global maximum points of $h$ by Ekeland's variational principle.

This paper serves as an errata for the article "P. W. Eloe, J. Henderson, J. Neugebauer, Electron. J. Qual. Theory Differ. Equ. 2020, No. 74, 1–15." In particular, the proof the authors give in that paper of Theorem 3.6 is incorrect, and so, that alleged theorem remains a conjecture. In this erratum, the authors state and prove a correct theorem.

We reprove Proposition 3.8 in our paper that was published in [Electron. J. Qual. Theory Differ. Equ. 2021, No. 7, 1–24], to fill a gap in the proof of Corollary 3.7 where the denseness of one of the embeddings does not follow by the original arguments. We further carry out some minor corrections in the proof of Corollary 3.7, in Remark 3.1 and in the formula (3.23) of the original paper.

In this paper, we consider the following fractional equation with a gradient term $$(-\Delta)^{s} u(x)= f(x,u (x), \nabla u (x)),$$ in a bounded domain and the upper half space. Firstly, we prove the monotonicity and uniqueness of solutions to the fractional equation in a bounded domain by the sliding method. In order to obtain maximum principle on unbounded domain, we need to estimate the singular integrals define the fractional Laplacians along a sequence of approximate maximum points by using a generalized average inequality. Then we prove monotonicity and uniqueness of solutions to fractional equation in $\mathbb{R}^n_+$ by the sliding method. In order to solve the difficulties caused by the gradient term, some new techniques are developed. The paper may be considered as an extension of Berestycki and Nirenberg [J. Geom. Phys. 5(1988), 237–275].

In this paper we give a new sufficient condition in order that all nontrivial Kneser solutions of the quasilinear ordinary differential equation \[ D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = (-1)^{n}p(t)|x|^{\beta}\mathrm{sgn}\,x, \quad t \geq a, \tag{1.1} \label{abseq} \] are singular. Here, $D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)$ is the $n$th-order iterated differential operator such that \begin{equation*} D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = D(\alpha_n)D(\alpha_{n-1})\cdots D(\alpha_1)x \end{equation*} and, in general, $D(\alpha)$ is the first-order differential operator defined by $D(\alpha)x = (d/dt)\left(|x|^{\alpha}\mathrm{sgn}\,x\right)$ for $\alpha > 0$. In the equation \eqref{abseq}, the condition $\alpha_1\alpha_2\cdots\alpha_n > \beta$ is assumed. If $\alpha_1 = \alpha_2 = \cdots = \alpha_n = 1$, then one of the results of this paper yields a well-known theorem of Kiguradze and Chanturia.

This paper deals with the blow-up of the solution for a system of evolution $p$-Laplacian equations $u_{it}=\text{div}(|\nabla u_{i}|^{p-2}\nabla u_{i})\;(i=1,2,\dots,k)$ with nonlinear boundary flux. Under certain conditions on the nonlinearities and data, it is shown that blow-up will occur at some finite time. Moreover, when blow-up does occur, we obtain the upper and lower bounds for the blow-up time. This paper generalizes the previous results.

In this paper, we study a hydrodynamic system modeling the evolution of a plasma subject to a self-induced electromagnetic Lorentz force in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a Maxwell equation. In the three dimensional case, we show that every weak solution verifies the energy equality for the incompressible Navier–Stokes–Maxwell equations with damping. We also establish some non-explosion criteria in terms of the velocity and magnetic of local strong solution for standard Navier–Stokes–Maxwell system.

In the present paper, we study the nonexistence of nontrivial weak solutions to a class of fractional $p$-Laplacian equation in two cases which are $sp > N$ and $sp < N$. In each of these cases, by using fractional Laplacian theory and inequality techniques, we obtain concrete range of parameter for which nontrivial weak solution of the problem does not exist. Our work complements the known nonexistence results in this direction.

In this paper, we use variational methods to prove the existence of a positive solution for the following class of logarithmic fractional Schrödinger–Poisson system: \begin{equation*} \begin{cases} \epsilon^{2s}\left(-\Delta\right)^{s} u+V(x)u-\phi(x)u= u \log {u^{2}}&\quad\text{ in }\mathbb{R}^{3}, \\ \epsilon^{2t}\left(-\Delta\right)^{t}\phi=|u|^{2}&\quad\text{ in }\mathbb{R}^{3}, \end{cases} \end{equation*} where $\epsilon>0$, $s,t\in(0,1)$, $\left(-\Delta\right)^{\alpha}$ is the fractional Laplacian and $V$ is a saddle-like potential.

In this paper, we study the existence of nodal solutions of some nonlinear boundary value problems for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. To do this, we first study the global bifurcation of solutions from zero and infinity of the corresponding nonlinear eigenvalue problems in classes with a fixed oscillation count. Then, using these global bifurcation results, we prove the existence of solutions of the considered nonlinear boundary value problems with a fixed number of nodes.

This paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set $\mathcal M$ in a continuously differentiable planar vector field by further characterizing for any point $p\in \mathcal M$, the composition of the limit sets $\omega (p)$ and $\alpha(p)$ after counting separately the fixed points on $\mathcal M$'s boundary and interior. In particular, when $\mathcal M$ contains finitely many boundary but no interior fixed points, $\omega (p)$ contains only a single fixed point, and when $\mathcal M$ may have infinitely many boundary but no interior fixed points, $\omega (p)$ can, in addition, be a continuum of fixed points. When $\mathcal M$ contains only one interior and finitely many boundary fixed points, $\omega (p)$ or $\alpha (p)$ contains exclusively a fixed point, a closed orbit or the union of the interior fixed point and homoclinic orbits joining it to itself. When $\mathcal M$ contains in general a finite number of fixed points and neither $\omega (p)$ nor $\alpha (p)$ is a closed orbit or contains just a fixed point, at least one of $\omega (p)$ and $\alpha (p)$ excludes all boundary fixed points and consists only of a number of the interior fixed points and orbits connecting them.

In this paper, we study the following Schrödinger–Hardy system \begin{equation*} \begin{cases} -\Delta_{\mathbb{G}}u-\mu\frac{\psi^2}{r(\xi)^2}u=F_u(\xi,u,v)\ &{\rm in}\ \Omega, \\ -\Delta_{\mathbb{G}}v-\nu\frac{\psi^2 }{r(\xi)^2}v=F_v(\xi,u,v)\ &{\rm in}\ \Omega, \\ u=v=0 \ \ & {\rm on}\ \partial\Omega, \end{cases} \end{equation*} where $\Omega $ is a smooth bounded domain on Carnot groups $\mathbb{G}$, whose homogeneous dimension is $Q\geq 3$, $\Delta_{\mathbb{G}}$ denotes the sub-Laplacian operator on $\mathbb{G}$, $\mu$ and $\nu$ are real parameters, $r(\xi)$ is the natural gauge associated with fundamental solution of $-\Delta_{\mathbb{G}}$ on $\mathbb{G}$, $\psi$ is the geometrical function defined as $\psi=|\nabla_{\mathbb{G}}r|$, and $\nabla_{\mathbb{G}}$ is the horizontal gradient associated with $\Delta_{\mathbb{G}}$. The difficulty is not only the nonlinearities $F_u$ and $F_v$ without Ambrosetti–Rabinowitz condition, but also the Hardy terms and the structure on Carnot groups. We obtain the existence of nonnegative solution for this system by mountain pass theorem in a new framework.

In this paper we study the well-posedness for the periodic Cauchy problem and the internal controllability of a one-dimensional system that describes the propagation of long water waves with small amplitude in the presence of surface tension. The well-posedness is proved by using the Fourier transform restriction method and the controllability is proved by using the moment method.

In this paper, we study a generalized Schrödinger–Poisson system in a bounded domain of $\mathbb{R}^{3}$ and involving an asymptotically linear nonlinearity. We prove the existence of positive solutions using bifurcation theory.

Consider the class ${\bf QS}$ of all non-degenerate planar quadratic differential systems and its subclass ${\bf QSP}$ of all its systems possessing an invariant parabola. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of ${\mathbb R}^{12}$ through its coefficients. In this paper, we provide necessary and sufficient conditions for a system in ${\bf QS}$ to have at least one invariant parabola. We give the global “bifurcation” diagram of the family ${\bf QS}$ which indicates where a parabola is present or absent and in case it is present, the diagram indicates how many parabolas there could be, their reciprocal position and what kind of singular points at infinity (simple or multiple) as well as their multiplicities are the points at infinity of the parabolas. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters.

This paper deals with a singular, nonlinear Sturm–Liouville problem of the form $\{A(x)u'(x)\}'+\lambda u(x)=f(x,u(x),u'(x))$ on $(0,1)$ where $A$ is positive on $(0,1]$ but decays quadratically to zero as $x$ approaches zero. This is the lowest level of degeneracy for which the problem exhibits behaviour radically different from the regular case. In this paper earlier results on the existence of bifurcation points are extended to yield global information about connected components of solutions.

The objective of this paper is to clarify the relationship between the $C^1$-smooth dependence of solutions to delay differential equations (DDEs) on initial histories (i.e., initial conditions) and delay parameters. For this purpose, we consider a class of DDEs which include a constant discrete delay. The problem of $C^1$-smooth dependence is fundamental from the viewpoint of the theory of differential equations. However, the above mentioned relationship is not obvious because the corresponding functional differential equations have the less regularity with respect to the delay parameter. In this paper, we prove that the $C^1$-smooth dependence on initial histories and delay holds by adopting spaces of initial histories of Sobolev type, where the differentiability of translation in $L^p$ plays an important role.

This paper is concerned with wave equations defined in domains of $\mathbb R^2$ with an invariable left boundary and a space-like right boundary which means the right endpoint is moving faster than the characteristic. Different from the case where the endpoint moves slower than the characteristic, this problem with ordinary boundary formulations may cause ill-posedness. In this paper, we propose a new kind of boundary condition to make systems well-posed, based on an idea of transposition. The key is to prove wellposedness and a hidden regularity for the corresponding backward system. Moreover, we establish an exponential decay estimate for the energy of homogeneous systems.

This paper is concerned with the existence of positive solutions for asymptotically periodic quasilinear Schr\"{o}dinger equations. By using a Nehari-type constraint and Moser iteration, we get the existence results which is a complement to the ones in Chu and Liu [Nonlinear Anal. Real World Appl. 44(2018), 118–127]. Moreover, we consider a new reformative asymptotic processes of the potential function and the nonlinearity term is only locally defined.

This paper deals with the oscillation problems for nonlinear differential equations of the form $(r(t)|x'|^{p(t)-2}x')'+c(t)f(x)=0$ involving $p(t)$-Laplacian. The Leighton–Wintner type oscillation criteria are established without any conditions on the limit of $p(t)$. In addition, we discuss the applications to partial differential equations. Some examples are given to illustrate our results.

This paper examines a class of fractional p-Kirchhoff systems driven by a nonlocal integro-differential operator with singular nonlinearity. By making use of Nehari manifold techniques, the existence of two nontrivial solutions is established. Our results extend those in Xiang et al. [Nonlinearity 29(2016), 3186–3205] for the corresponding subcritical case.

In the present paper, we further study the asymptotical behavior of the following higher order nonlinear difference equation \begin{equation*} x(n+1)= ax(n)+ bf( x(n)) + cf(x(n-k)), \qquad n=0, 1, \dots \end{equation*} where $a, b $ and $c$ are constants with $00$ for $x>0$, and $k$ is a positive integer, which has been recently studied in: On global attractivity of a higher order difference equation and its applications [Electron. J. Qual. Theory Diff. Equ. 2022, No. 2, 1–14]. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation, and show the applications of these results to some population models.

In this paper we define the notion of slow divergence integral along sliding segments in regularized planar piecewise smooth systems. The boundary of such segments may contain diverse tangency points. We show that the slow divergence integral is invariant under smooth equivalences. This is a natural generalization of the notion of slow divergence integral along normally hyperbolic portions of curve of singularities in smooth planar slow-fast systems. We give an interesting application of the integral in a model with visible-invisible two-fold of type $VI_3$. It is related to a connection between so-called Minkowski dimension of bounded and monotone "entry-exit" sequences and the number of sliding limit cycles produced by so-called canard cycles.

In this paper, we consider the radially symmetric spacelike solutions of a nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime. By using a conformal change of variable, this problem can be translated an equivalent problem in the Minkowski spacetime. By using the lower and upper solution method, fixed point, a priori bounds and topological degree method, we obtain the existence, nonexistence and multiplicity of radially symmetric spacelike solutions.

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals $\Phi(u)=\int_a^b f(x,u(x),u'(x))\,dx$, where $u\in C^1([a,b],{\mathbb R}^N)$. We first derive conditions which are simpler than the known ones, and then apply them to several particular problems, including stability problems in the elasticity theory. In particular, we solve some open problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and buckling solutions in three dimensions, Nonlinearity 27(2014), 2841–2867] by finding optimal conditions for the stability of a naturally straight Kirchhoff rod under various types of endpoint constraints.

Stieltjes differential equations, i.e. differential equations with usual derivatives replaced by derivatives with respect to given functions (derivators), are useful to model processes which exhibit dead times and/or sudden changes. These advantages of Stieltjes equations are exploited in this paper in the analysis of two real life models: first, the frictionless motion of a vehicle equipped with an electric engine and, second, the evolution of populations of cyanobacteria Spirullina plantensis in semicontinuous cultivation processes. Furthermore, this is not only a paper on applications of known results. For the adequate analysis of our mathematical models we first deduce the solution formula for Stieltjes equations with separate variables. Finally, we show that differential equations with Stieltjes derivatives reduce to ODEs when the derivator is continuous, thus obtaining another resolution method for more general cases.

In this paper, we study the existence of multi-bump solutions for the following Schrödinger–Bopp–Podolsky system with steep potential well: \begin{equation*} \begin{cases} -\Delta u+(\lambda V(x)+V_0(x))u+K(x)\phi u= |u|^{p-2}u, &x\in \mathbb{R}^3,\\ -\Delta \phi+a^2\Delta^2\phi=K(x) u^2, &x\in \mathbb{R}^3, \end{cases} \end{equation*} where $p \in(4,6), a>0$ and $\lambda$ is a parameter. We require that $V(x) \geq 0$ and has a bounded potential well $\Omega=V^{-1}(0)$. Combining this with other suitable assumptions on $\Omega, V_{0}$ and $K$, when $\lambda$ is large enough, we obtain the existence of multi-bump-type solutions $u_{\lambda}$ by using variational methods.

In this paper, we devote ourselves to considering a modified zero energy critical point theory for a specific set of functionals denoted as $\Phi_{\mu}$, defined within the confines of a uniformly convex Banach space. Integrating the nonlinear generalized Rayleigh quotient approach with Ljusternik–Schnirelman category, we establish the nonexistence and multiplicity of zero energy critical points of the involved functionals. In particular, the modified zero energy critical point theory can be applied to more nonlocal problems. Our main results improve and complement the existing results in the related literature.

This paper is devoted to studying the existence of positive solutions of the problem: \begin{equation} \begin{cases}\label{0.1}\tag{$\ast$} -\Delta u=\frac{u^{p}}{|x|^{a}}+h(x,u,\nabla u), & \mbox{in} \ \Omega,\\ u=0, & \mbox{on}\ \partial\Omega,\\ \end{cases} \end{equation} where $\Omega\subset \mathbb{R}^{N}(N\geq3)$ is an open bounded smooth domain with boundary $\partial\Omega$, and $1

A coupled damped Klein–Gordon–Schrödinger equations are considered where $\Omega$ is a bounded domain of $\mathbb{R}^{2},$ with smooth boundary $\Gamma$ and $\omega$ is a neighbourhood of $\partial \Omega$ satisfying the geometric control condition. The aim of the paper is to prove the existence, uniqueness and uniform decay for the solutions.

An exact solution of an initial value problem for the Susceptible–Exposed–Infectious–Recovered–Deceased (SEIRD) epidemic model is derived, and various properties of the exact solution are obtained. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation of the second kind. In this paper Abel differential equations play an important role in establishing the exact solution of the SEIRD differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an initial value problem for an Abel differential equation. Uniqueness of positive solutions of an initial value problem to SEIRD differential system is also investigated, and it is shown that the exact solution is a unique solution in the class of positive solutions.

In this paper, we consider of the following second-order Hamiltonian system \begin{equation*} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\qquad \forall t \in \mathbb{R}, \end{equation*} where $W(t,x)$ is subquadratic at infinity. With a competition condition, we establish the existence of homoclinic solutions by using the variational methods. In our theorem, the smallest eigenvalue function $l(t)$ of $L(t)$ is not necessarily coercive or bounded from above and $W(t,x)$ is not necessarily integrable on $\mathbb{R}$ with respect to $t$. Our theorem generalizes many known results in the references.

In this paper, we study the following fractional Kirchhoff type equation \begin{equation*} \begin{cases} \left(a+b\displaystyle\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_p^su=|u|^{q-2}u\ln |u|^2+\frac{\lambda}{u^\gamma}, &\rm \mathrm{in}\ \Omega, \\ u>0, &\rm \mathrm{in}\ \Omega, \\ u=0, &\rm \mathrm{in}\ \mathbb{R}^N\backslash \Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $00, b\geq0$, $N>ps$, $2p0$ is a real parameter. By using the critical point theory for nonsmooth functionals and analytic techniques, the existence and multiplicity of positive solutions are obtained.

This paper is concerned with a predator-prey model with cannibalism and prey-evasion. The global existence and boundedness of solutions to the system in bounded domains of 1D and 2D are proved for any prey-evasion sensitivity coefficient. It is also shown that prey-evasion driven Turing instability when the prey-evasion coefficient surpasses the critical value. Besides, the existence of Hopf bifurcation, which generates spatiotemporal patterns, is established. And, numerical simulations demonstrate the complex dynamic behavior.

In this paper, we study the bifurcation of limit cycles from a class of cubic integrable non-Hamiltonian systems under arbitrarily small piecewise smooth perturbations of degree $n$. By using the averaging theory and complex method, the lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems are given at first order in $\varepsilon$. It is also shown that in this case, the maximum number of limit cycles produced by piecewise smooth perturbations is almost twice the upper bound of the maximum number of limit cycles produced by smooth perturbations for the considered systems.

In this paper, we consider the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a fourth-order equation with strong damping and logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the method of Faedo–Galerkin approximation. By means of potential well method and Rellich inequality, we obtain the global existence and the decay estimate of global solutions under some appropriate conditions. Furthermore, we prove the finite time blow-up results of weak solutions, and establish the upper and lower bounds for blow-up time.