The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter $\lambda$ under Robin boundary conditions in unbounded domains, with (possibly noncompact) smooth boundary. The problem involves a weighted $p$-Laplacian operator and subcritical nonlinearities, and even in the case $p=2$ the main existence results are new. Denoting by $\lambda_1$ the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case $\lambda\ge \lambda_1$ or $\lambda <\lambda_1$. In the first part of the paper we show the existence of a nontrivial solution for all $\lambda\in\mathbb R$ for the problem under Ambrosetti--Rabinowitz-type conditions on the nonlinearities involved in the model. In detail, we apply the mountain-pass theorem of Ambrosetti and Rabinowitzif $\lambda <\lambda_1$, while we use mini-max methods and linking structures over cones, as in Degiovanni [10]and in Degiovanni and Lancelotti [11], if $\lambda\ge \lambda_1$. In the latter part of the paper we do not require any longer the Ambrosetti--Rabinowitz condition at $\infty$, but the so-called Szulkin--Weth conditions, and we obtain the same result for all $\lambda\in\mathbb R$. More precisely, using the Nehari-manifold method for $C^1$ functionals developed by Szulkin and Weth in [38], we prove existence of ground states, multiple solutions, and least-energy sign-changing solutions, whenever $\lambda <\lambda_1$. On the other hand, in the case $\lambda\ge \lambda_1$, we establish the existence of solutions again by linking methods.

We consider the Cauchy problem for dispersion-managed nonlinear Schrödinger equations, where the dispersion map is assumed to be periodic and piecewise constant in time. We establish local and global well-posedness results and the possibility of finite time blow-up. In addition, we shall study the scaling limit of fast dispersion management and establish convergence to an effective model with averaged dispersion.

The main purpose of the paper is to study the large-time behavior of positive solutions of a semilinear elliptic equation with a dynamical boundary condition. We show that small solutions behave asymptotically like suitable multiples of the Poisson kernel.

We consider the Navier--Stokes equations for viscous incompressible flows in the half plane under the no-slip boundary condition. In this paper we first establish a solution formula for the vorticity equations through the appropriate vorticity formulation. The formula is then applied to establish the asymptotic expansion of vorticity fields at $\nu\rightarrow 0$ that holds at least up to the time $c\nu^{1/3}$, where $\nu$ is the viscosity coefficient and $c$ is a constant. As a consequence, we get a natural sufficient condition on the initial data for the vorticity to blow up at the inviscid limit, together with explicit estimates.

In this paper, we consider the borderline solution to the semilinear equations with critical growth. A concentration phenomenon of the solution when the time goes to infinity is proved. First, we show that a $\varepsilon$-regularity property holds for an $H^1$ solution to the related elliptic equation, and then give a precise description of the formation of the bubbles. A similar bubbling description is also derived for the harmonic maps on surface. (Cf.~Struwe [22], Qing [19], Qing-Tian [20], Chen-Tian [6], Lin-Wang [13], and Parker [16]).

We completely determine the range of Sobolev regularity for the Maxwell--Dirac system in $1+1$ space time dimensions to be well-posed locally in the case that the initial data of the Dirac part regularity is of $L^2$. The well-posedness follows from the standard energy estimates. Outside the range for the well-posedness, we show either the flow map is not continuous or not twice differentiable at zero.