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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 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.

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 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.

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.