We study the nonlinear Schrödinger type equation $$-\Delta u~+~(\lambda g(x)~+~1)u~=~f(u)$$ on the whole space ${{\mathbf R}^N}$. The nonlinearity $f$ is assumed to be asymptotically linear and $g(x)\geq 0$ has a potential well. We do not assume a limit for $g(x)$ as $|x|\to\infty$. Using variational techniques, we prove the existence of a positive solution for $\lambda$ large. In the case where $f$ is odd we obtain multiple pairs of solutions. The limiting behavior of solutions as $\lambda\to\infty$ is also considered.

Bifurcation results for some problems with different kinds of lack of compactness are discussed. The techniques used are, mainly, topological and variational.

The purpose of this article is to study the exact boundary controllability of the classical Boussinesq equation. The control is applied to the first spatial derivative and then, to the second spatial derivative, both at the right endpoint. The exact controllability of the linearized problem is essentially proved by using the Hilbert Uniqueness Method. From this result, we deduce the exact boundary controllability for the nonlinear Boussinesq equations. The main improvements compared to existing results in the literature are the absence of restrictions on controllability time, the use of more regular spaces and the extension of the exact boundary controllability to the nonlinear Boussinesq equation.

We prove the equivalence of the well-posedness of a partial differential equation with nonautonomous past and an associated abstract Cauchy problem. This equivalence is then used to derive sufficient conditions for exponential stability of the solutions.

In this work we consider the following class of nonlinear perturbed Schrödinger equations $$ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^{2}}{2m}\Delta \psi+V(x,z)\psi-\gamma|\psi|^{p-2}\psi ,$$$\gamma, $ $m>0,$$(x,z)=(x_1,x_2,z)\in{{\mathbb R}}^{3}$, where $\hbar>0$, $2 <p <1+\sqrt{5}$,$\psi:{{\mathbb R}}^{3}\rightarrow{{\mathbb C}}.$ Here, the potential $V$ is bounded from below away from zero and satisfies: $V(x,z)=V(|x|,z)$ for all $(x,z)=(x_1,x_2,z)\in{{\mathbb R}}^{3}$. We are interested in finding solutions having the form $\psi(r,z,\theta,t)=\exp\left[ i(\omega\theta-Et)/\hbar\right]v(r,z),$ being $(r,z,\theta)$ the cylindrical coordinates in the space. When the parameter $\hbar$ approaches zero our solutions concentrate around a circle lying on a plane $z=\overline{z}$ with center $(0,0,\overline{z})$.