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Results in Journal Electronic Journal of Qualitative Theory of Differential Equations: 1,450

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Lingju Kong,
Electronic Journal of Qualitative Theory of Differential Equations pp 1-9; https://doi.org/10.14232/ejqtde.2021.1.60

Abstract:
In this article, we investigate the existence of positive solutions of a boundary value problem for a system of fractional differential equations. The resilience of a fractional compartment system is also studied to demonstrate the application of the result.
Yingying Xiao, Chuanxi Zhu
Electronic Journal of Qualitative Theory of Differential Equations pp 1-17; https://doi.org/10.14232/ejqtde.2021.1.73

Abstract:
In this paper, we study the following quasilinear Schrödinger equation − Δ u + V ( x ) u − κ u Δ ( u 2 ) + μ h 2 ( | x | ) | x | 2 ( 1 + κ u 2 ) u + μ ( ∫ | x | + ∞ h ( s ) s ( 2 + κ u 2 ( s ) ) u 2 ( s ) d s ) u = f ( u ) in R 2 , κ > 0 V ∈ C 1 ( R 2 , R ) and f ∈ C ( R , R ) By using a constraint minimization of Pohožaev–Nehari type and analytic techniques, we obtain the existence of ground state solutions.
Mingliang Song, Runzhen Li
Electronic Journal of Qualitative Theory of Differential Equations pp 1-13; https://doi.org/10.14232/ejqtde.2021.1.68

Abstract:
We obtain an existence theorem of nonzero solution for a class of bounded selfadjoint operator equations. The main result contains as a special case the existence result of a nontrivial homoclinic orbit of a class of Hamiltonian systems by Coti Zelati, Ekeland and Séré. We also investigate the existence of nontrivial homoclinic orbit of indefinite second order systems as another application of the theorem.
Zijian Wu, Haibo Chen
Electronic Journal of Qualitative Theory of Differential Equations pp 1-16; https://doi.org/10.14232/ejqtde.2021.1.71

Abstract:
In this article, we study the multiplicity of solutions for a class of fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$. Under the appropriate assumption, we prove that there are at least two solutions for the equation by Nehari manifold and Ekeland variational principle, one of which is the ground state solution.
, Jeidy Jimenez,
Electronic Journal of Qualitative Theory of Differential Equations pp 1-38; https://doi.org/10.14232/ejqtde.2021.1.69

Abstract:
Due to their applications to many physical phenomena during these last decades the interest for studying the discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system, but to determine the maximum number of limits cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of the planar differential systems. Thus in general to provide a sharp upper bound for the number of crossing limit cycles that a given class of piecewise linear differential system can have is a very difficult problem. In this paper we characterize the existence and the number of limit cycles for the piecewise linear differential systems formed by linear Hamiltonian systems without equilibria and separated by a reducible cubic curve, formed either by an ellipse and a straight line, or by a parabola and a straight line parallel to the tangent at the vertex of the parabola. Hence we have solved the extended 16th Hilbert problem to this class of piecewise differential systems.
Xiaojing Feng
Electronic Journal of Qualitative Theory of Differential Equations pp 1-29; https://doi.org/10.14232/ejqtde.2021.1.59

Abstract:
In this paper, we obtain the existence of positive critical point with least energy for a class of functionals involving nonlocal and supercritical variable exponent nonlinearities by applying the variational method and approximation techniques. We apply our results to the supercritical Schrödinger–Poisson type systems and supercritical Kirchhoff type equations with variable exponent, respectively.
Zhongxiang Wang,
Electronic Journal of Qualitative Theory of Differential Equations pp 1-18; https://doi.org/10.14232/ejqtde.2021.1.83

Abstract:
The paper focuses on the modified Kirchhoff equation  \begin{align*} -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)\Delta u-u\Delta (u^2)+V(x)u=\lambda f(u), \quad x\in \mathbb{R}^N, \end{align*} where $a,b>0$, $V(x)\in C(\mathbb{R}^N,\mathbb{R})$ and $\lambda<1$ is a positive parameter. We just assume that the nonlinearity $f(t)$ is continuous and superlinear in a neighborhood of $t = 0$ and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. Then we use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. Moreover, the nonlinearity f(t) may be supercritical.
Jan Jekl
Electronic Journal of Qualitative Theory of Differential Equations pp 1-17; https://doi.org/10.14232/ejqtde.2021.1.79

Abstract:
In this paper, we investigate even-order linear difference equations and their criticality. However, we restrict our attention only to several special cases of the general Sturm–Liouville equation. We wish to investigate on such cases a possible converse of a known theorem. This theorem holds for second-order equations as an equivalence; however, only one implication is known for even-order equations. First, we show the converse in a sense for one term equations. Later, we show an upper bound on criticality for equations with nonnegative coefficients as well. Finally, we extend the criticality of the second-order linear self-adjoint equation for the class of equations with interlacing indices. In this way, we can obtain concrete examples aiding us with our investigation.
Tohid Kasbi, Vahid Roomi, Aliasghar Jodayree Akbarfam
Electronic Journal of Qualitative Theory of Differential Equations pp 1-13; https://doi.org/10.14232/ejqtde.2021.1.34

Electronic Journal of Qualitative Theory of Differential Equations pp 1-6; https://doi.org/10.14232/ejqtde.2021.1.80

Abstract:
We extend the planar Markus–Yamabe Jacobian conjecture to differential systems having Jacobian matrix with eigenvalues with negative or zero real parts.
Electronic Journal of Qualitative Theory of Differential Equations pp 1-9; https://doi.org/10.14232/ejqtde.2021.1.77

Abstract:
In the research literature, one can find distinct notions for higher order averaged functions of regularly perturbed non-autonomous T-periodic differential equations of the kind x ′ = ε F ( t , x , ε ) . By one hand, the classical (stroboscopic) averaging method provides asymptotic estimates for its solutions in terms of some uniquely defined functions gi's, called averaged functions, which are obtained through near-identity stroboscopic transformations and by solving homological equations. On the other hand, a Melnikov procedure is employed to obtain bifurcation functions fi's which controls in some sense the existence of isolated T-periodic solutions of the differential equation above. In the research literature, the bifurcation functions fi's are sometimes likewise called averaged functions, nevertheless, they also receive the name of Poincaré–Pontryagin–Melnikov functions or just Melnikov functions. While it is known that f 1 = T g 1 , a general relationship between gi and fi is not known so far for i ≥ 2 . Here, such a general relationship between these two distinct notions of averaged functions is provided, which allows the computation of the stroboscopic averaged functions of any order avoiding the necessity of dealing with near-identity transformations and homological equations. In addition, an Appendix is provided with implemented Mathematica algorithms for computing both higher order averaging functions.
, Yiqing Li
Electronic Journal of Qualitative Theory of Differential Equations pp 1-18; https://doi.org/10.14232/ejqtde.2021.1.85

Abstract:
In this paper, we dedicate to studying the following semilinear Schrödinger system e q u a t i o n * - Δ u + V 1 ( x ) u = F u ( x , u , v ) a m p ; m b o x i n ~ R N , r - Δ v + V 2 ( x ) v = F v ( x , u , v ) a m p ; m b o x i n ~ R N , r u , v ∈ H 1 ( R N ) , e n d e q u a t i o n * where the potential Vi are periodic in x,i=1,2, the nonlinearity F is allowed super-quadratic at some x ∈ R N and asymptotically quadratic at the other x ∈ R N . Under a local super-quadratic condition of F, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.
Leila Gholizadeh,
Electronic Journal of Qualitative Theory of Differential Equations pp 1-8; https://doi.org/10.14232/ejqtde.2021.1.78

Abstract:
We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a turning point in the principal part of the equation. Related results are discussed.
Yonghui Tong, Hui Guo,
Electronic Journal of Qualitative Theory of Differential Equations pp 1-14; https://doi.org/10.14232/ejqtde.2021.1.70

Abstract:
We consider a class of fractional logarithmic Schrödinger equation in bounded domains. First, by means of the constraint variational method, quantitative deformation lemma and some new inequalities, the positive ground state solutions and ground state sign-changing solutions are obtained. These inequalities are derived from the special properties of fractional logarithmic equations and are critical for us to obtain our main results. Moreover, we show that the energy of any sign-changing solution is strictly larger than twice the ground state energy. Finally, we obtain that the equation has infinitely many nontrivial solutions. Our result complements the existing ones to fractional Schrödinger problems when the nonlinearity is sign-changing and satisfies neither the monotonicity condition nor Ambrosetti-Rabinowitz condition.
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