Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term

In this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev's inequality and Wirtinger's inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic case, we obtain two existence results of weak quasi-periodic solutions for the second order Hamiltonian system: $$\frac{d[P(t)\dot{u}(t)]}{dt}=\nabla F(t,u(t))+ e(t),$$ which generalize and improve the corresponding results in recent literature [J. Kuang, Abstr. Appl. Anal. 2012, Art. ID 271616]. Moreover, when the assumptions $F(t,x)=F(t,-x)$ and $e(t)\equiv 0$ are also made, we obtain two results on existence of infinitely many weak quasi-periodic solutions for the second order Hamiltonian system under the subquadratic case.

Periodic solutions for second-order Hamiltonian systems with a p-Laplacian

In this paper, by using the least action principle, Sobolev’s inequality and Wirtinger’s inequality, some existence theorems are obtained for periodic solutions of second-order Hamiltonian systems with a p-Laplacian under subconvex condition, sublinear growth condition and linear growth condition. Our results generalize and improve those in the literature

Existence of solutions for a poly-Laplacian system involving concave-convex nonlinearity on locally finite graphs

We investigate the existence of two nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearity and parameters with Dirichlet boundary value condition on the locally finite graph. By using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one non-semi-trivial solution of positive energy and one non-semi-trivial solution of negative energy, respectively. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result in [10] which is based on the fibering maps and the method of Nehari manifold, we obtain the existence of ground state solution to the single equation corresponding to poly-Laplacian system. Especially, we present the concrete range of parameters in all of results

Ground state sign-changing solutions for second order elliptic equation with logarithmic nonlinearity on locally finite graphs

We obtain the existence results of ground state sign-changing solutions and ground state solutions for a class of second order elliptic equation with logarithmic nonlinearity on a locally finite graph $G=(V,E)$, and obtain the sign-changing ground state energy is larger than twice of the ground state energy. The method we used is a direct non-Nehari manifold method in [X.H. Tang, B.T. Cheng. J. Differ. Equations. 261(2016), 2384-2402.

Ground state sign-changing homoclinic solutions for a discrete nonlinear pp-Laplacian equation with logarithmic nonlinearity

By using a direct non-Nehari manifold method from [X.H. Tang, B.T. Cheng. J. Differ. Equations. 261(2016), 2384-2402.], we obtain an existence result of ground state sign-changing homoclinic solution which only changes sign one times and ground state homoclinic solution for a class of discrete nonlinear $p$-Laplacian equation with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground state energy is larger than twice of the ground state energy

Ground state solutions for a non-local type problem in fractional Orlicz Sobolev spaces

In this paper, we study the following nonlocal problem in fractional Orlicz Sobolev spaces \begin{eqnarray*} (-\Delta_{\Phi})^{s}u+V(x)a(|u|)u=f(x,u),\quad x\in\mathbb{R}^N, \end{eqnarray*} where $(-\Delta_{\Phi})^{s}(s\in(0, 1))$ denotes the non-local and maybe non-homogeneous operator, the so-called fractional $\Phi$-Laplacian. Without assuming the Ambrosetti-Rabinowitz type and the Nehari type conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case. The proof is based on a variant version of the mountain pass theorem and a Lions' type result for fractional Orlicz Sobolev spaces

An equilibrium-conserving taxation scheme for income from capital

Under conditions of market equilibrium, the distribution of capital income follows a Pareto power law, with an exponent that characterizes the given equilibrium. Here, a simple taxation scheme is proposed such that the post-tax capital income distribution remains an equilibrium distribution, albeit with a different exponent. This taxation scheme is shown to be progressive, and its parameters can be simply derived from (i) the total amount of tax that will be levied, (ii) the threshold selected above which capital income will be taxed and (iii) the total amount of capital income. The latter can be obtained either by using Piketty's estimates of the capital/labor income ratio or by fitting the initial Pareto exponent. Both ways moreover provide a check on the amount of declared income from capital.Comment: 4 pages, 2 figure

Existence of weak quasi-periodic solutions for a second order Hamiltonian system with damped term via a PDE approach

In this paper, we investigate the existence of weak quasi-periodic solutions for the second order Hamiltonian system with damped term: \begin{equation} \ddot{u}(t)+q(t)\dot{u}(t)+D W(u(t))=0, \qquad t\in \mathbb R,\tag{HSD} \end{equation} where $u:\mathbb R\to \mathbb R^n$, $q:\mathbb R\to \mathbb R$ is a quasi-periodic function, $W:\mathbb R^n\to \mathbb R$ is continuously differentiable, $DW$ denotes the gradient of $W$, $W(x)=-K(x)+F(x)+H(x)$ for all $x\in \mathbb R^n$ and $W$ is concave and satisfies the Lipschitz condition. Under some reasonable assumptions on $q, K,F,H$, we obtain that system has at least one weak quasi-periodic solution. Motivated by Berger et al. (1995) and Blot (2009), we transform the problem of seeking a weak quasi-periodic solution of system (HSD) into a problem of seeking a weak solution of some partial differential system. We construct the variational functional which corresponds to the partial differential system and then by using the least action principle, we obtain the partial differential system has at least one weak solution. Moreover, we present two propositions which are related to the working space and the variational functional, respectively