We consider the Cauchy problem for some semi-linear wave equations in three space dimensions and prove global or almost global existence of the Klainerman-Machedon radial solutions. The proof is carried out by a contraction-mapping argument based on a refined version of the Keel-Smith-Sogge estimate, together with the Morawetz-type inequality.

We prove a bifurcation theorem of Crandall-Rabinowitz type (local bifurcation of smooth families of nontrivial solutions) for general variational inequalities on possibly nonconvex sets with infinite-dimensional bifurcation parameter. The result is based on local equivalence of the inequality to a smooth equation with Lagrange multipliers, on scaling techniques and on an application of the implicit function theorem. As an example, we consider a semilinear elliptic PDE with nonconvex unilateral integral conditions on the boundary of the domain.

In a recent paper [3] we proved global well posedness for the Dirac-Klein-Gordon equations with Yukawa interaction in $1+1$ dimensions with initial data $\psi_{0}\in L^2(\mathbb R) $, $(\phi_{0},\phi_{1})\in {H^{r}(\mathbb R)} \times {H^{r-1}(\mathbb R)} $, where $r\in[1/4,1]$, $r\neq 1/2$. In this paper we use a new null form estimate to prove global well posedness for the more difficult case $r=1/2$.

We are concerned with the uniform positivity preserving property on a domain $D$ of $\mathbb{R}^d$ ($d\geq 3$), for the noncooperative system \begin{equation}\label{sy} \left\{ \begin{array}{cccc} -\Delta u & = & f(.,u)-\mu av & \text{in } D, \\ -\Delta v & = & bu & \text{in }D, \\ \underset{ x \rightarrow \partial_{\infty} D }{\lim }u(x) & = & \underset{ x \rightarrow \partial_{\infty} D}{\lim }v(x) & = 0, \end{array} \right. \end{equation} where $\partial_{\infty}D=\left\{ \begin{array}{ccc} \partial D ,\ \ \mbox{if D is bounded},\\ \partial D\cup \{+\infty\}, \ \ \mbox{if not}. \end{array} \right.$ We give appropriate conditions on $a$, $b$ and $f$ to get the existence and positivity of the solutions with potential control.

In this paper we study the first (nonlinear) Steklov eigenvalue, $\lambda$, of the following problem: $-\Delta_{p}u + |u|^{p-2}u + \alpha\phi|u|^{p-2}u = 0$ in a bounded smooth domain $\Omega$ with $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu} = \lambda |u|^{p-2}u$ on the boundary $ \partial\Omega$. We analyze the dependence of this first eigenvalue with respect to the weight $\phi$ and with respect to the parameter $\alpha$. We prove that for fixed $\alpha$ there exists an optimal $\phi_\alpha$ that minimizes $\lambda$ in the class of uniformly bounded measurable functions with fixed integral. Next, we study the limit of these minima as the parameter $\alpha$ goes to infinity and we find that the limit is the first Steklov eigenvalue in the domain with a hole where the eigenfunctions vanish.

In this paper, we study the large-time behavior of solutions to the Schrödinger equation with the quadratic nonlinearity $\lambda |u|^2$ with $\lambda \in \mathbb{C} \setminus \{ 0 \}$ in two space dimensions. The nonexistence of non-trivial scattering states for this equation is proved without assuming a convergence rate for the difference between a solution to the nonlinear equation and a free solution.

In this paper we study the existence, regularity and geometric properties of an optimal configuration to a free boundary optimization problem governed by the $p$-Laplacian operator.