The existence of at least one classical T-periodic solution is proved for differential equations of the form \begin{eqnarray*} (\phi(u'))' - g(x,u) = h(x) \end{eqnarray*} when $\phi : (-a,a) \to {\mathbb R}$ is an increasing homeomorphism, $g$ is a Carathéodory function T-periodic with respect to $x$, $2\pi$-periodic with respect to $u$, of mean value zero with respect to $u,$ and $h \in L^1_{loc}(R)$ is T-periodic and has mean value zero. The problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space of T-periodic Lipschitz functions, and then to show, using variational inequalities techniques, that such a minimum solves the differential equation. A special case is the ``relativistic forced pendulum equation'' \begin{eqnarray*} \Big (\frac{u'}{\sqrt{1 - u'^2}} \Big )' + A \sin u = h(x). \end{eqnarray*}

Let $A=-(\nabla-i\vec{a})^2+V$ be a magnetic Schrödinger operator acting on $L^2(\mathbb{R}^n), n\geq 1$, where $\vec{a}=(a_1,\ldots, a_n)\in L^2_{loc}$ and $0\leq V \in L^1_{ loc}$. In this paper we will give weighted $L^p$ estimates for Riesz transforms $(\partial/\partial x_k -ia_k )A^{-1/2}$ associated with $A, k=1, \ldots, n,$ and their commutators with an appropriate range of $p$. Note that our obtained results extend those in [11, 12].

Let ${u_{{\varepsilon}}}$ be a least energy solution to the nearly critical problem \[ -{\Delta} u = c_0 K(x) u^{{p_{{\varepsilon}}}} \; \mbox{in} \; \Omega, \quad u>0 \; \mbox{in} \; \Omega, \quad u|_{{\partial} \Omega} = 0, \] where $\Omega \subset {\mathbb{R}}^N (N \ge 3)$ is a smooth bounded domain, $c_0 = N(N-2)$, ${p_{{\varepsilon}}} = (N+2)/(N-2) - {\varepsilon}$ where ${\varepsilon}>0$ is a small parameter and $K \in C^2(\overline{\Omega})$ is a positive function. Under some assumptions on $K$, we prove several asymptotic estimates of the eigenvalues ${\lambda}_{i,{\varepsilon}}$ and corresponding eigenfunctions $v_{i,{\varepsilon}}$ to the eigenvalue problem \begin{align*} \begin{cases} -{\Delta} v_{i,{\varepsilon}} = {\lambda}_{i,{\varepsilon}} \left( c_0 {p_{{\varepsilon}}} K(x) {u_{{\varepsilon}}}^{{p_{{\varepsilon}}}-1} \right) v_{i,{\varepsilon}} & \mbox{in} \; \Omega, \\ v_{i,{\varepsilon}} = 0 & \mbox{on} \; {\partial}\Omega, \\ \| v_{i,{\varepsilon}} \|_{L^{\infty}(\Omega)} = 1 & \end{cases} \end{align*} as ${\varepsilon} \to 0$, for $i= 2, \cdots, N+1, N+2$.

We prove the existence of planar closed curves with prescribed curvature $ \kappa $ ($\kappa $-loops) for classes of symmetric curvature functions $\kappa :\mathbb{C}\rightarrow \mathbb{R}$ of any sign, either exhibiting some homogeneity, or satisfying a uniform condition on the growth along radial directions. The problem of $\kappa $-loops is equivalent to the problem of $1$-periodic solutions $u\in C^{2}(\mathbb{R},\mathbb{C})$ to a nonlinear ODE, namely $u^{\prime \prime }=i\left\| u\right\| _{L^{2}(\left[ 0,1\right] )}\kappa \left( u\right) u^{\prime }$, which also bears different physical and geometrical interpretations. Such a problem is variational in nature and, thanks to low dimension, the main difficulty is the existence of bounded Palais-Smale sequences, which cannot be granted by standard arguments of critical point theory.

This work is devoted to proving the global existence of weak solutions for a general isothermal model of capillary fluids derived in his modern form by C. Rohde in [21], which can be used as a phase transition model. First, inspired by the result by P.-L. Lions in [19] on the Navier-Stokes compressible system we show the global stability of weak solutions for our system with isentropic pressure and next with general pressure. Finally, we consider perturbations of a stable equilibrium.

A general class of linear and nonautonomous delay differential equations with initial data in a separable Hilbert space is treated. The classic questions of existence, uniqueness, and regularity of solutions are addressed. Moreover, the semigroup approach typically adopted in the autonomous case for continuous initial functions is extended, and thus the existence of an equivalent abstract ordinary formulation is shown to hold. Finally, the existence of infinitely many Lyapunov exponents for the associated evolution is proven and their meaning is discussed.

In this paper we consider structurally damped plate and wave equations with point and distributed random forces. In order to treat space dimensions more than one, we work in the setting of $L^q$--spaces with (possibly small) $q\in(1,2)$. We establish existence, uniqueness and regularity of mild and weak solutions to the stochastic equations employing recent theory for stochastic evolution equations in UMD Banach spaces.

The aim of this paper is to establish a higher integrability result for the second derivatives of solutions to nondivergence elliptic equations of the type $\sum_{i,j}^n a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} =h$. The matrix coefficient $A(x)=[a_{ij}(x)]_{ij}$ is assumed to belong to the space $BMO$.