This paper is concerned with the existence of saddle type solutions for a class of semilinear elliptic equations of the type \begin{equation} \Delta u(x)+F_{u}(x,u)=0,\quad x\in\mathbb R^{n},\;\; n\ge 2, \tag*{(PDE)} \end{equation} where $F$ is a periodic and symmetric nonlinearity. Under a non degeneracy condition on the set of minimal periodic solutions, saddle type solutions of $(PDE)$ are found by a renormalized variational procedure.

This work is concerned with the Cauchy problem for a coupled Schrödinger-Benjamin-Ono system \begin{equation*} \left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=\alpha uv, \hfill t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+\nu\mathcal H\partial^2_xv=\beta \partial_x(|u|^2),\\ u(0,x)=\phi, \ v(0,x)=\psi, \ \ \ \ \ \ \ \ \ \ \ \ \ \hfill (\phi,\psi)\!\in\!H^{s}(\mathbb R)\!\times\!H^{s'}\!(\mathbb R). \end{array} \right. \end{equation*} In the ${\it non-resonant}$ case $(|\nu|\ne1)$, we prove local well-posedness for a large class of initial data. This improves the results obtained by Bekiranov, Ogawa and Ponce (1998). Moreover, we prove $C^2$-{\it ill-posedness} at ${\it low-regularity}$, and also when the difference of regularity between the initial data is large enough. As far as we know, this last ill-posedness result is the first of this kind for a nonlinear dispersive system. Finally, we also prove that the local well-posedness result obtained by Pecher (2006) in the ${\it resonant}$ case $(|\nu|=1)$ is sharp except for the end-point.

Our goal in this paper is to investigate the effect of Fourier law on a new coupling for the Bresse system. We give an explicit characterization on the decay rate that can be exponential or polynomial type, depending on the relation between the coefficients of the wave speed propagation.

We address the global regularity of solutions to the Boussinesq equations with zero diffusivity in two spatial dimensions. Previously, the persistence in the space $H^{1+s}(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2)$ for all $s\ge 0$ has been obtained. In this paper, we address the persistence in general Sobolev spaces, establishing it on a time interval which is almost independent of the size of the initial data. Namely, we prove that if $(u_0,\rho_0)\in W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $s\in(0,1)$ and $q\in[2,\infty)$, then the solution $(u(t),\rho(t))$ of the Boussinesq system stays in $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $t\in[0,T^*)$, where $T^*$ depends logarithmically on the size of initial data. If we furthermore assume that $sq>2$, then we get the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$. Moreover, we prove the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for the initial data with compact support, as well us for data in $W^{1+s,q}(\mathbb{T}^2)\times W^{s,q}(\mathbb{T}^2)$, without any restriction on $s\in(0,1)$ and $q\in[2,\infty)$.

Existence and uniqueness of local strong solutions for the Beris--Edwards model for nematic liquid crystals, which couples the Navier-Stokes equations with an evolution equation for the $Q$-tensor, is established on a bounded domain $\Omega\subset\mathbb{R}^d$ in the case of homogeneous Dirichlet boundary conditions. The classical Beris--Edwards model is enriched by including a dependence of the fluid viscosity on the $Q$-tensor. The proof is based on a linearization of the system and Banach's fixed-point theorem.

We prove an epiperimetric inequality for the thin obstacle problem; thus, extending the pioneering results by Weiss on the classical obstacle problem ({Invent. Math.}, 138 (1999), 23--50). This inequality provides the means to study the rate of converge of the rescaled solutions to their limits, as well as the regularity properties of the free boundary.