We investigate some boundary-value problems for an Emden--Fowler-type differential system \[ u_{1}^{\prime}=g_{1}(t)u_{2}^{\lambda_{1}},\;\;\;u_{2}^{\prime}=g_{2} (t)u_{1}^{\lambda_{2}} \] on a finite or infinite interval $I=[a,b)$, where $g_{i}:I\rightarrow \lbrack0,\infty)$ $(i=1,2)$ are locally integrable functions. We give the optimal, in a certain sense, sufficient conditions that guarantee the existence of a unique (at least of one) nonnegative solution, satisfying one of the two following boundary conditions: \[ \mathrm{i)}\ u_{1}(a)=c_{0},\;\;\;\lim_{t\rightarrow b}u_{1}(t)=c;\;\; \mathrm{ii)}\ u_{2}(a)=c_{0},\;\;\;\lim_{t\rightarrow b}u_{1}(t)=c, \] in case $0\leq c_{0} < c < +\infty$ (in case $c_{0}\geq0$, $c=+\infty$ and $\lambda_{1}\lambda_{2}>1$). Moreover, the global two-sided estimations of the above-mentioned solutions are obtained together with applications to differential equations with $p$-Laplacian.

We establish some properties for the Cauchy problem associated with the nonlocal nonlinear Schrödinger equation $\partial_{t}u=-i\alpha\partial^{2}_{x}u+\beta u\partial_{x}(\,|u|^{2})-i\beta u\mathcal{T}_{h}\partial_{x}(\,|u|^{2}) +i\gamma \left| u\right| ^{2}u,$ where $x, t\;\in\mathbb R,$ $\mathcal{T}_{h}$ is the nonlocal operator $$ \mathcal{T}_{h}u(x)=\frac{1}{2h}p.v. \int^{\infty}_{-\infty}\coth \Big (\frac{\pi(y-x)}{2h} \Big ) u(y)dy, $$ with $\alpha>0$, $\beta\geq 0$, $\gamma \geq 0$, and $h\in(0,+\infty)$. Here $\mathcal{T}_{h}\longrightarrow\mathcal{H}$ when $h\longrightarrow +\infty,$ where $\mathcal{H}$ is the Hilbert transform. We prove rigorously that a Picard interaction scheme can not be applied for solving the Cauchy problem associated with that equation in both the cases $0 < h < \infty$ and $h\rightarrow +\infty,$ with initial data in Sobolev spaces of negative index. Elsewhere, we study the asymptotic behavior of the solution in relation to a spatial variable, and we also establish the nonexistence of a standing-waves solution for the above equation in several cases.

We consider the problem $- \Delta u+V(x)u=f^{\prime}(u)+g(x)$ in $\mathbb R^N$, under the assumption $\lim_{x\rightarrow\infty}V(x)=0$, and with the nonlinear term $f$ with a double power behavior. We prove the existence two solutions when $g$ is sufficiently small and $V < 0$.

We record some Bernstein properties for entire solutions to higher-order elliptic equations, including certain bi-Laplacian equations, other equations which are second order in the eigenvalues of the Hessian of the solution and some geometrically inspired fourth-order equations. We apply a Bernstein result for second-order elliptic equations on noncompact Riemannian manifolds, whose proof is based on techniques of Yau [20].

We prove that the equation modelling an annular Josephson junction has a rotating fluxon wave solution for all values of the parameters. We also obtain results on uniqueness of the rotating fluxon wave in some parameter regimes, and on multiplicity of rotating fluxon waves in other parameter regimes.

In this paper, we study the long-time asymptotics of the Cauchy problem for the Hamilton-Jacobi equation $$ u_t(x,t) + \alpha x \cdot Du(x,t) + H(Du(x,t)) = f(x) \ \ \text{ in } \ \ {\mathbb R }^n \times (0,\infty) , $$ where $\alpha$ is a positive constant. In \cite{FIL2}, it was shown that there are a constant $c \in {\mathbb R }$ and a viscosity solution $v$ of $c + \alpha x\cdot Dv(x) + H(Dv(x)) = f(x)$ in ${\mathbb R }^n$ such that $u(\cdot,t) -(v(\cdot)+ct) \to 0$ as $t \to \infty$ locally uniformly in ${\mathbb R }^n$. The function $v(x) + ct$ is called the asymptotic solution. Our goal is to give a sufficient condition in order that the set of points where the rate of this convergence is slower than $t^{-1}$ is non-empty. We also give several examples which show that we can not remove, in general, the assumptions in this sufficient condition in order that this set is non-empty. As a result, we clarify crucial factors which cause this slow rate of convergence. They are both a geometrical property of the set of equilibrium points and a lower bound of the initial data.