Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems

AbstractThe purpose of this paper is two-fold. Firstly, we will give some parabolic-like conditions which improve the well-known angle conditions and allow further computations of the critical groups both at degenerate critical points and at infinity. As an application, we then consider the second-order Hamiltonian systemsu″(t)+∇H(t,u(t))=0,t∈R, where H:R×RN→R is T-periodic in its first variable and ∇H is asymptotically linear both at origin and at infinity. Based on the computations of the critical groups and the Morse theory, we obtain the existence and multiplicity results for periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature

Non-existence and multiplicity of positive solutions for Choquard equations with critical combined nonlinearities

We study the non-existence and multiplicity of positive solutions of the nonlinear Choquard type equation $$-\Delta u+ \varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u, \quad {\rm in} \ \mathbb R^N, \qquad (P_\varepsilon)$$ where $N\ge 3$ is an integer, $p\in [\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}]$, $q\in (2,\frac{2N}{N-2}]$, $I_\alpha$ is the Riesz potential of order $\alpha\in (0,N)$ and $\varepsilon>0$ is a parameter. We fix one of $p,q$ as a critical exponent (in the sense of Hardy-Littlewood-Sobolev and Sobolev inequalities ) and view the others in $p,q,\varepsilon, \alpha$ as parameters, we find regions in the $(p,q,\alpha, \varepsilon)$-parameter space, such that the corresponding equation has no positive ground state or admits multiple positive solutions. This is a counterpart of the Brezis-Nirenberg Conjecture (Brezis-Nirenberg, CPAM, 1983) for nonlocal elliptic equation in the whole space. Particularly, some threshold results for the existence of ground states and some conditions which insure two positive solutions are obtained. These results are quite different in nature from the corresponding local equation with combined powers nonlinearity and reveal the special influence of the nonlocal term. To the best of our knowledge, the only two papers concerning the multiplicity of positive solutions of elliptic equations with critical growth nonlinearity are given by Atkinson, Peletier (Nonlinear Anal, 1986) for elliptic equation on a ball and Juncheng Wei, Yuanze Wu (Proc. Royal Soc. Edinburgh, 2022) for elliptic equation with a combined powers nonlinearity in the whole space. The ODE technique is main ingredient in the proofs of the above mentioned papers, however, ODE technique does not work any more in our model equation due to the presence of the nonlocal term.Comment: 55pages. arXiv admin note: text overlap with arXiv:2302.1372

Normalized solutions for the Choquard equation with mass supercritical nonlinearity

We consider the nonlinear Choquard equation $$\begin{cases} & - \Delta u = (I_\alpha \ast F(u))F'(u) -\mu u \ \text{in}\ \mathbb{R}^N, & u \in \ H^1(\mathbb{R}^N), \ \int_{\mathbb{R}^N} |u|^2 dx=m, \end{cases}$$ where $\alpha\in(0,N)$, $m>0$ is prescribed, $\mu \in \mathbb{R}$ is a Lagarange multiplier, and $I_\alpha$ is the Riesz potential. Under general assumptions on the nonlinearity $F,$ we prove the existence and multiplicity of normalized solutions.Comment: arXiv admin note: text overlap with arXiv:2002.03973 by other author

Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$-\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N,$$ as $\lambda\to 0$ and $\lambda\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\ge 0$ are constants and $\lambda>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $\lambda\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-\Delta u+u=u^{q-1}$ and as $\lambda\to +\infty$, after another rescaling the ground state solutions of the problem converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such rescalings, which depends in a non-trivial way on the space dimension $N=3$ and $N= 4$. We also discuss a connection of our results with a mass constrained problem associated to the Kirchhoff equation with the mass normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$.Comment: 40 page

Propagation and its failure in a lattice delayed differential equation with global interaction

AbstractWe study the existence, uniqueness, global asymptotic stability and propagation failure of traveling wave fronts in a lattice delayed differential equation with global interaction for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. In the bistable case, under realistic assumptions on the birth function, we prove that the equation admits a strictly monotone increasing traveling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotic stable with phase shift. Of particular interest is the phenomenon of “propagation failure” or “pinning” (that is, wave speed c = 0), we also give some criteria for pinning in this paper

Semilinear Duffing Equations Crossing Resonance Points

AbstractIn this paper, using a generalized form of the Poincaré–Birkhoff theorem and a fixed point theorem, we prove, under weaker conditions, two theorems for the equationx+g(x)=p(t),p(t)≡p(t+2π), of which one shows the existence of a harmonic solution, the other that the equation may have an infinite number of harmonic solutions in the resonance case. This is an enhancement of the results already obtained

Traveling Wave Solutions for Planar Lattice Differential Systems with Applications to Neural Networks

AbstractWe obtain some existence results for traveling wave fronts and slowly oscillatory spatially periodic traveling waves of planar lattice differential systems with delay. Our approach is via Schauder's fixed-point theorem for the existence of traveling wave fronts and via S1-degree and equivarant bifurcation theory for the existence of periodic traveling waves. As examples, the obtained abstract results will be applied to a model arising from neural networks and explicit conditions for traveling wave fronts and global continuation of periodic waves will be obtained

COMPARATIVE STUDY OF ON-SITE SORTING FOR C&amp;D IN CHINA AND EUROPE

Construction and demolition waste (CDW) accounts for 40% of urban municipal waste in China and around 25% in the European Union (EU). Since the EU is more developed and urbanized than China, its experience with managing CDW may be helpful to China. This study therefore compared China and the EU with respect to the flow of CDW materials and the policies, laws and regulations for CDW management. The results reveal that the CDW management practices and facilities in China are relatively underdeveloped with a large amount of low-value inert material going to landfill compared with the EU. The study also reveals the important role of government involvement in CDW management, including the use of punitive measures and preferential policies; most EU members states achieved their waste recovery rates by 2016 due to mature CDW legalization. To improve the management of CDW in China, a series of suggestions are proposed including waste prevention strategies, establishment of supervision mechanisms, and financial support. </jats:p