Narrow Results

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form $\frac{d}{dt}(\nabla \mathrm{\Phi }(\dot{u}(t)))+V_u(t,u(t))+\lambda G_u(t,u(t))=0$, where $\mathrm{\Phi }:{ℝ}^n\to [0,\infty )$ is a ${𝒢}$-function in the sense of Trudinger, $V,G:ℝ\times {ℝ}^n\to ℝ$ are $C^1$-smooth, $T$-periodic in the time variable $t$ and $\lambda$ is a real parameter. We prove the existence of a $T$-periodic solution $u_{\lambda }:ℝ\to {ℝ}^n$ for any sufficiently small $\left|\lambda \right|$, and show that the found solutions converge to a $T$-periodic solution of the unperturbed system if $\lambda$ tends to $0$. Let us stress that our theorem only makes a natural assumption on the potential $V$ and no additional assumption on $G$ except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges.

In this work, we study second-order Hamiltonian systems under small perturbations. We assume that the main term of the system has a mountain pass structure, but do not suppose any condition on the perturbation. We prove the existence of a periodic solution. Moreover, we show that periodic solutions of perturbed systems converge to periodic solutions of the unperturbed systems if the perturbation tends to zero. The assumption on the potential that guarantees the mountain pass geometry of the corresponding action functional is of independent interest as it is more general than those by Rabinowitz [Homoclinic orbits for a class of Hamiltonian systems, Proc. R. Soc. Edinburgh A 114 (1990) 33–38] and the authors [M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second-order Hamiltonian systems, J. Differ. Equ. 219 (2005) 375–389].

We prove the existence of infinitely many non-radial positive solutions for the Schrödinger–Newton system

We present some of our results concerning the existence of multiple solutions to elliptic differential equations. In particular, we deal with the Dirichlet problem involving the p-Laplacian and the periodic solutions to second order Hamiltonian systems. In all of these results, we follow a variational approach. We look for solutions of the considered problem which are in turn local minima for the underlying energy functional.