We consider an incompressible, viscous, finite depth fluid flowing down a three-dimensional channel. In the absence of surface tension, we prove the existence of a unique stationary solution in weighted Sobolev spaces. The result is based on a thorough study of the linearized problem, particularly the pseudodifferential operator relating the normal velocity of the fluid and the normal component of the associated stress tensor along the free surface, and requires the use of the Nash-Moser implicit function theorem.

This work deals with a mathematical analysis of an age-cycle structured population endowed with a transition rate and with a general biological rule. We show that the corresponding model is governed by a strongly continuous positive semigroup. Some of its spectral properties are proved.

This paper is concerned with the multiplicity of nontrivial solutions to the class of nonlocal boundary value problems of the Kirchhoff type, \begin{equation*} - \Big [M \Big (\int_{\Omega}|\nabla u|^{2} \ dx\ \Big ) \Big ]\Delta u = \lambda |u|^{q-2}u+|u|^{p-2}u \ \mbox{in} \ \Omega, \ \ \mbox{and} \ u=0 \ \mbox{on} \ \ \partial\Omega, \end{equation*} where $\Omega\subset\mathbb R^{N}$, for $N=1,$ 2, and 3, is a bounded smooth domain, $1 < q < 2 < p \leq 2^{*}=6$ in the case $N=3$ and $2^{*}=\infty$ in the case $N=1$ or $N=2$. Our approach is based on the genus theory introduced by Krasnoselskii [22].

We study a Neumann problem for a nonlinear elliptic system. Unlike previous results in the literature of Landesman--Lazer type, our existence theorem allows rapid rotations on the nonlinear term.

We study the global Cauchy problem associated to the Davey-Stewartson system in ${\mathbb{R}}^n,\ n=2,3$. Existence and uniqueness of the solution are established for small data in some weak $L^p$ space. We apply an interpolation theorem and the generalization of the Strichartz estimates for the Schrödinger equation derived in [9]. As a consequence we obtain self-similar solutions.

We consider the two-parameter nonlinear eigenvalue problem $$ -u''(t) = \mu u(t) - \lambda(u(t) + u(t)^p), \ u(t) > 0, t \in I := (0, 1), \ \ u(0) = u(1) = 0, $$ where $p > 1$ is a constant and $\mu, \lambda > 0$ are parameters. We establish the precise asymptotic formulas for the variational eigencurve $\lambda = \lambda(\mu)$, which is defined on a general level set, as $\mu \to \infty$. Especially, we obtain new critical exponents $p = 7/5$, $p = 5/3$, $p = 2$, and $p = 5$ from the viewpoint of the asymptotics of the eigencurve $\lambda(\mu)$.

We develop virial theorems for ordinary differential equations. Both the Finkelstein and Fock methods are used. Vanishing of boundary conditions at singular points are handled by asymptotic methods as well as conditions which guarantee that the singular point is both a strong limit point and Dirichlet. Applications are given to determine intervals which contain no eigenvalues.

We consider the Cauchy problem in ${\bf R}^{n}$ for wave equations with a fractional damping. We generalize partially the previous results due to[12], and derive sharp decay estimates for the total energy and the $L^{2}$-norm of solutions based on the energy method in the Fourier space.

This paper is concerned with a fractional PDE that interpolates semilinear heat and wave equations. We show results on global-in-time well-posedness for small initial data in the critical Morrey spaces and space dimension $n\geq1$. We also remark how to derive the local-in-time version of the results. Qualitative properties of solutions like self-similarity, antisymmetry and positivity are also investigated. Moreover, we analyze the asymptotic stability of the solutions and obtain a class of asymptotically self-similar solutions.

In this paper, by means of the extraction of the Palais--Smale sequence in the Nehari manifold, we are concerned with the existence of multiple positive solutions of a class of the p-Laplacian equations involving concave-convex nonlinearities $$\left\{ \begin{array}{ll} -\triangle_p u+|u|^{p-2}u=a(x)|u|^{s-2}u +\lambda b(x)|u|^{r-2}u,\;\;\; x\in {\mathbb R}^N,\\ u\in W^{1,p}({\mathbb R}^N), \end{array} \right.$$ in the whole space ${\mathbb R}^N,$ where $\lambda$ is a positive constant, $1\leq r < p < s < p^*=\frac{Np}{N-p}$, and $a(x)$ and $b(x)$ are nonnegative continuous functions in ${\mathbb R}^N.$