We introduce a method to treat a semilinear elliptic equation in ${ℝ}^N$ (see equation (1) below). This method is of a perturbative nature. It permits us to skip the problem of lack of compactness of ${ℝ}^N$ but requires an oscillatory behavior of the potential b.

This paper is concerned with the following periodic Hamiltonian elliptic system $\{-\Delta \varphi +V\left(x\right)\varphi =G_{\psi }(x,\varphi ,\psi )\text{in}{ℝ}^N,-\Delta \psi +V\left(x\right)\psi =G_{\varphi }(x,\varphi ,\psi )\text{in}{ℝ}^N,\varphi \left(x\right)\to 0\text{and}\psi \left(x\right)\to 0\text{as}|x|\to \infty .$ Assuming the potential V is periodic and 0 lies in a gap of $\sigma (-\Delta +V)$, $G(x,\eta )$ is periodic in x and asymptotically quadratic in $\eta =(\varphi ,\psi )$, existence and multiplicity of solutions are obtained via variational approach.


We consider the existence of infinitely many solutions to the boundary value problem $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left({\frac{1}{2}}_0{D}_t^{-\beta }\left(u^{\text{'}}\left(t\right)\right)+{\frac{1}{2}}_t{D}_T^{-\beta }\left(u^{\text{'}}\left(t\right)\right)\right)+\nabla F(t,u\left(t\right))=0\text{a.e.}t\in [0,T],\\ u\left(0\right)=u\left(T\right)=0.\end{array}$$ Under more general assumptions on the nonlinearity, we obtain new criteria to guarantee that this boundary value problem has infinitely many solutions in the superquadratic, subquadratic and asymptotically quadratic cases by using the critical point theory.

Under no Ambrosetti-Rabinowitz-type growth condition, we study the existence of infinitely many solutions of the p(x)-Laplacian equations by applying the variant fountain theorems due to Zou [Manuscripta Math. 104 (2001), 343-358].

Under a suitable oscillatory behavior either at infinity or at zero of the nonlinear term, the existence of infinitely many weak solutions for a non-homogeneous Neumann problem, in an appropriate Orlicz--Sobolev setting, is proved. The technical approach is based on variational methods.

We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy...

We analyse the influence of diffusion and space distribution of cells in a simple model of interactions between an activated immune system and malignant gliomas, among which the most aggressive one is GBM Glioblastoma Multiforme. It turns out that diffusion cannot affect stability of spatially homogeneous steady states. This suggests that there are two possible outcomes-the solution is either attracted by the positive steady state or by the semitrivial one. The semitrivial steady state describes...

Propagation of polymerization fronts with liquid monomer and liquid polymer is considered and the influence of vibrations on critical conditions of convective instability is studied. The model includes the heat equation, the equation for the concentration and the Navier-Stokes equations considered under the Boussinesq approximation. Linear stability analysis of the problem is fulfilled, and the convective instability boundary is found depending on...

The aim of this paper is to study the effect of vibrations on convective instability of reaction fronts in porous media. The model contains reaction-diffusion equations coupled with the Darcy equation. Linear stability analysis is carried out and the convective instability boundary is found. The results are compared with direct numerical simulations.

We present families of scalar nonconforming finite elements of arbitrary order $r\ge 1$ with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order $r-1$ form inf-sup stable finite element pairs of order r for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case r=1. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order...

Ingham [6] ha migliorato un risultato precedente di Wiener [23] sulle serie di Fourier non armoniche. Modificando la sua funzione di peso noi otteniamo risultati ottimali, migliorando precedenti teoremi di Kahane [9], Castro e Zuazua [3], Jaffard, Tucsnak e Zuazua [7] e di Ullrich [21]. Applichiamo poi questi risultati a problemi di osservabilità simultanea.