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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 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. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

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∞-stable and asymptotically stable. Besides, an illustrative example is provided to support the theoretical results. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fractional equation with a gradient term (-Δ)su(x) = f(x,u), Δu(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 Rn+ 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. [ABSTRACT FROM AUTHOR]

In this paper we give a new sufficient condition in order that all nontrivial Kneser solutions of the quasilinear ordinary differential equation... and, in general, D(α) is the first-order differential operator defined by D(α)x = (d/dt) (|x|αsgn x) for a > 0. In the equation (1.1), the condition α1&#94652 ... αn > β is assumed. If α1 = α2 = ... = αn = 1, then one of the results of this paper yields a well-known theorem of Kiguradze and Chanturia. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up of the solution for a system of evolution p-Laplacian equations uit = div(|Δui|p-2Δui) (i = 1, 2, . . ., 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. [ABSTRACT FROM AUTHOR]

This paper deals with a singular, nonlinear Sturm-Liouville problem of the form fA(x)u0(x)g0 +lu(x) = f (x, u(x), u0(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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with the stabilization of the linear biharmonic Schrödinger equation in an n-dimensional open bounded domain under Dirichlet-Neumann boundary conditions considering three infinite memory terms as damping mechanisms. We show that depending on the smoothness of initial data and the arbitrary growth at infinity of the kernel function, this class of solution goes to zero with a polynomial decay rate like t-n depending on assumptions about the kernel function associated with the infinite memory terms. [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasilinear Schrödinger-Poisson system in R³ ∫ -Δu + V(x)u + λϕu = f (x, u), x ∈ R³, -Δϕ - ε4Δ4ϕ = λu², x ∊ R³, where λ and ε are positive parameters, Δ4u = div(|∇u|²∇u), V is a continuous and periodic potential function with positive infimum, f (x, t) ∊ C(R³ × R,R) is periodic with respect to x and only needs to satisfy some superquadratic growth conditions with respect to t. One nontrivial solution is obtained for λ small enough and ε fixed by a combination of variational methods and truncation technique. [ABSTRACT FROM AUTHOR]

In this paper, we prove the structural stability for a family of scalar reactiondiffusion equations. Our arguments consist of using invariant manifold theorem to reduce the problem to a finite dimension and then, we use the structural stability of Morse-Smale flows in a finite dimension to obtain the corresponding result in infinite dimension. As a consequence, we obtain the optimal rate of convergence of the attractors and estimate the Gromov-Hausdorff distance of the attractors using continuous e-isometries. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Kirchhoff type equation ∫(a + b ∫RN NRN |u(x) - u(y)|p/|x - y|N+ps dxdy) (-Δ)spu = |u|q-2u ln |u|² + λ/uγ, in Ω, u > 0, in Ω, u = 0, in RN\Ω, where Ω ⊂ RN is a bounded domain with Lipschitz boundary, 0 < s < 1 < p, 0 < γ < 1, a > 0, b ≥ 0, N > ps, 2p < q < q + 2 < p*s, p*s = Np/N-ps is the fractional critical exponent, λ > 0 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. [ABSTRACT FROM AUTHOR]

Direct spring operated pressure relief valves connected to a constantly charged vessel and a downstream pipe have a complex dynamics. The vessel-valve subsystem is described with an autonomous system of ordinary differential equations, while the presence of the pipe adds two partial differential equations to the mathematical model. The partial differential equations are transformed to a delay algebraic equation coupled to the ordinary differential equations. Due to a square root nonlinearity, the system is implicit. The linearized system can be transformed to a standard system of neutral delay differential equations (NDDEs) having more elaborated literature than the delay algebraic equations. First, the different forms of the mathematical model are presented, then the transformation of the linearized system is conducted. The paper aims at introducing this unusual mathematical model of an engineering system and inducing research focusing on the methodology to carry out bifurcation analysis in implicit NDDEs. [ABSTRACT FROM AUTHOR]

In this paper, we obtain three subharmonic solutions with different prescribed minimal periods for a class of second order impulsive differential systems. Our proof is based on level estimates and the least action principle in critical point theory. [ABSTRACT FROM AUTHOR]

In this paper, we offer new technique for investigation of the even order linear differential equations of the form y(n)(t) = p(t)y(τ(t)). (E) We establish new criteria for bounded and unbounded oscillation of (E) which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicities for the positive solutions of (E) than those presented in known works. We illustrate the improvement over known results by applying and comparing our technique with the other known methods on the particular examples. [ABSTRACT FROM AUTHOR]

In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the m(x)-Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace-Beltrami operator on the unit sphere. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of autonomous nonlinear oscillators with non-autonomous first integral. We prove explicitly the existence of a global sink which is, under some conditions, an algebraic limit cycle. For that class, we draw the possible phase portraits in the Poincaré disk. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents: ∫-Δu = |u|2∗-2u + α/2* |u|α-2|v|βu + λ|u|q-2u, in Ω, -Δv = |v|2*-2v + β/2* |u|α|v|β-2v + μ|v|q-2v, in Ω, u, v > 0, in Ω, u = v = 0, on ∂Ω, where Ω ⊂ RN(N ≥ 3) is a bounded domain with smooth boundary, λ, μ > 0, 1 < q < 2, α > 1, β > 1, α + β = 2* = 2N/N-2. By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we look for solutions to the following critical Schrödinger system ∫-Δu + (V1 + λ1) u = |u|2* - 2 u + |u|p1-2 u + βr1|u|r1-2 u|v|r2 in RN, - Δv + (V2 + λ2)v = |v|2* - ²v + |v|p2 - 2v + βr2|u|r1|v|r2-2 v in RN, having prescribed mass ∫RNu² = a1 > 0 and ∫RNv² = a2 > 0, where λ1, λ2 ∊ R will arise as Lagrange multipliers, N ⩾ 3, 2 * = 2 N/(N-2) is the Sobolev critical exponent, r1,r2 > 1, p1, p2, r1 + r2 ∊ (2 + 4/N, 2*) and β > 0 is a coupling constant. Under suitable conditions on the potentials V1 and V2, β* > 0 exists such that the above Schrödinger system admits a positive radial normalized solution when β ≤ β *. The proof is based on comparison argument and minmax method. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for some Λ-Ω differential systems to have an analytic commutator, use these properties to judge the origin point of the Λ-Ω differential systems to be an isochronous center. [ABSTRACT FROM AUTHOR]

In this paper, based on a suitable fractional Trudinger--Moser inequality, we establish sufficient conditions for the existence result of least energy sign-changing solution for a class of one-dimensional nonlocal equations involving logarithmic and exponential nonlinearities. By using a main tool of constrained minimization in Nehari manifold and a quantitative deformation lemma, we consider both subcritical and critical exponential growths. This work can be regarded as the complement for some results of the literature. [ABSTRACT FROM AUTHOR]

In this paper we investigate nonlinear systems of second order ODEs describing the dynamics of two coupled nonlinear oscillators of a mechanical system. We obtain, under certain assumptions, some stability results for the null solution. Also, we show that in the presence of a time-dependent external force, every solution starting from sufficiently small initial data and its derivative are bounded or go to zero as the time tends to +8, provided that suitable conditions are satisfied. Our theoretical results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

This paper deals with fundamental properties of Poincaré half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincaré half-maps, their series expansions (Taylor and Newton-Puiseux) at the tangency point and at infinity, the relative position between the graph of Poincaré halfmaps and the bisector of the fourth quadrant, and the sign of their second derivatives. All these properties are essential to understand the dynamic behavior of planar piecewise linear systems. Accordingly, we also provide some of their most immediate, but non-trivial, consequences regarding periodic orbits. [ABSTRACT FROM AUTHOR]

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 e. 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. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mixed boundary value problem for nonuniformly elliptic equation in a variable exponent Sobolev space containing p(·)-Laplacian and mean curvature operator. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of a nontrivial weak solution and at least two nontrivial weak solutions according to some hypotheses on given functions. [ABSTRACT FROM AUTHOR]

This paper discusses a class of nonlinear singular perturbation problems with Robin boundary values in critical cases. By using the boundary layer function method and successive approximation theory, the corresponding asymptotic expansions of small parameters are constructed, and the existence of uniformly efficient smooth solutions is proved. Meanwhile, we give a concrete example to prove the validity of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear Klein-Gordon-Maxwell system with a steep potential well ∫-Δu + (λa(x) + 1)u - μ(2ω + ϕ)ϕu = f (x, u), inR³, Δϕ = μ(ω + ϕ)u², inR³, where ω > 0 is a constant, μ and λ are positive parameters, f ∈ C(R3 × R,R) and the nonlinearity f satisfies the Ambrosetti-Rabinowitz condition. We use parameterdependent compactness lemma to prove the existence of nontrivial solution for μ small and λ large enough, then explore the asymptotic behavior as μ → 0 and λ → ∞. Moreover, we also use truncation technique to study the existence and asymptotic behavior of positive solutions of the Klein-Gordon-Maxwell system when f (u) := |u|q-2u where 2 < q < 4. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to continue earlier works by the authors on the oscillation problem of second-order half-linear neutral delay differential equations. By revising the set method, we present new oscillation criteria which essentially improve a number of related ones from the literature. A couple of examples illustrate the value of the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in general. Additionally, we also apply the aforementioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem. [ABSTRACT FROM AUTHOR]

In this paper, the Hopf bifurcations and Turing bifurcations of the Gierer-Meinhardt activator-inhibitor model are studied. The very interesting and complex spatially periodic solutions and patterns induced by bifurcations are analyzed from both theoretical and numerical aspects respectively. Firstly, the conditions for the existence of Hopf bifurcation and Turing bifurcation are established in turn. Then, the Turing instability region caused by diffusion is obtained. In addition, to uncover the diffusion mechanics of Turing patterns, the dynamic behaviors are studied near the Turing bifurcation by using weakly nonlinear analysis techniques, and the type of spatial pattern was predicted by the amplitude equation. And our results show that the spatial patterns in the Turing instability region change from the spot, spot-stripe to stripe in order. Finally, the results of the analysis are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the following quasilinear parabolic-ellipticelliptic chemotaxis system ∫ut = ∇·(ϕ(u)∇u - ψ(u)∇v) + au - buγ, x ∈ Ω, t > 0, 0 = ∆v - v + wγ1, x ∈ Ω, t > 0, 0 = ∆w - w + uγ2, x ∈ Ω, t > 0, with homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ Rn (n ≥ 1), where a, b, γ2 > 0, γ1 ≥ 1, γ > 1 and the functions ϕ, ψ ∈ C² ([0, ∞) satisfy ϕ(s) ≥ a0(s + 1)α and |ψ(s)| ≤ b0s(1 + s)β-1 for all s ≥ 0 with a0,b0 > 0 and α, β ∈ R. It is proved that if γ - β ≥ γ1γ2, the classical solution of system would be globally bounded. Furthermore, a specific model for γ1 = 1, γ2 = κ and γ = κ + 1 with κ > 0 is considered. If β ≤ 1 and b > 0 is large enough, there exist Cκ, µ1, µ2 > 0 such that the solution(u, v, w) satisfies..., for all t ≥ 0. The above results generalize some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity solutions for the following Klein-Gordon-Maxwell system ∫-Δu + V(x)u - (2ω + ϕ)ϕu = f (x, u), x ∈ R³, Δϕ = (ω + ϕ)u², x ∊ R³, where ω > 0 is a constant and the nonlinearity f (x, u) is either asymptotically linear in u at infinity or the primitive of f (x, u) is of 4-superlinear growth in u at infinity. Under some suitable assumptions, the existence and multiplicity of solutions are proved by using the Mountain Pass theorem and the fountain theorem, respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of age-structured cholera model with general infection rates. We first explore the existence and uniqueness, dissipativeness and persistence of the solutions, and the existence of the global attractor by verifying the asymptotical smoothness of the orbits. We then give mathematical analysis on the existence and local stability of the positive equilibrium. Based on the preparation, we further investigate the global behavior of the cholera infection model. Corresponding numerical simulations have been presented. Our results improve and generalize some known results on cholera models. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for the following generalized quasilinear Schrödinger equation - div(gp(u)|∇u|p-2∇u) + gp-1(u)g′(u)|∇u|p + V(x)|u|p-2u = K(x) f (u) + Q(x)g(u)|G(u)|p*-2G(u), x ∈ RN, where N ≥ 3, 1 < p ≤ N, p* = Np/N-p, g ∈ C¹(R,R+), V(x) and K(x) are positive continuous functions and G(u) =∫u0g(t)dt. By using a change of variable, we obtain the existence of positive solutions for this problem by using the Mountain Pass Theorem. Our results generalize some existing results. [ABSTRACT FROM AUTHOR]

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.