In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing-sign solutions. By means of the Cayley transform we will set the problem on the sphere $S^{2n+1}$; since the functional $I$ associated with the equation does not satisfy the Palais-Smale compactness condition, we will find a suitable closed subspace $X$ on which we can apply the minmax argument for $I_{|X}$. We generalize the result to any compact contact manifold of $K$-contact type.

It is shown that if $(u,v)$ is a sufficiently smooth solution of the initial value problem associated with the Schrödinger-Korteweg-de Vries system such that there exist $a, b\in\mathbb{R}$ with $\operatorname{supp}u(t_j)\subseteq(a,\infty)$ (or$(-\infty,a)$) and $\operatorname{supp}v(t_j)\subseteq(b,\infty)$ (or $(-\infty,b)$), for $j=1,2 \ (t_1\neq t_2)$, then $u\equiv v\equiv0$.

We prove interior $L^p$ gradient weighted estimates for degenerate elliptic equations in divergence form with VMO coefficients.

Let $\Omega$ be a $\mathcal{C}^{2}$ bounded domain of ${{\mathbb R}}^N$, $N\geq 2$. We consider the following quasilinear elliptic problem: $$ ({ P}_{\lambda})\left\lbrace \begin{array}{l} -\Delta_p u = K(x)(\lambda u^q-u^r),\quad \ \ \mbox{ in }\Omega, \\ \quad \;\;\;u= 0 \quad\mbox{ on }\partial\Omega, \quad u\geq 0\quad\mbox{ in }\Omega, \end{array}\right. $$ where $p>1$ and $\Delta_p u{\stackrel{{\rm {def}}}{=}} \mathrm{div} \left(\vert \nabla u\vert ^{p-2}\nabla u\right)$ denotes the $p$-Laplacian operator. In this paper, $\lambda>0$ is a real parameter, the exponents $q$ and $r$ satisfy $-1<r<q<p-1$, and $K:\Omega\longrightarrow {{\mathbb R}}$ is a positive function having a singular behaviour near the boundary $\partial \Omega$. Precisely, $K(x)=d(x)^{-k}L(d(x))$ in $\Omega$, with $0<k<p$, $L$ a positive perturbation function, and $d(x)$ the distance of $x\in\Omega$ to $\partial\Omega$. By using a sub- and supersolution technique, we discuss the existence of positive solutions or compact support solutions of $({ P}_{\lambda})$ in respect to the blow-up rate $k$. Precisely, we prove that if $k<1+r$, $({ P}_{\lambda})$ has at least one positive solution for $\lambda>0$ large enough, whereas it has only compact support solutions if $k\geq 1+r$.

We study the multiplicity of positive solutions of the problem $$ -\Delta_p u+|u|^{p-2}u=0 $$ in a bounded smooth domain $\Omega\subset{\mathbb{R}}^N$, with a nonlinear boundary condition given by $$ |\nabla u|^{p-2}\partial u/\partial\nu=\lambda f(u) +\mu\varphi(x)|u|^{q-1}u, $$ where $f$ is continuous and satisfies some kind of $p-$superlinear condition at 0 and $p-$sublinear condition at infinity, $0<q< p-1$ and $\varphi$ is $L^\beta(\partial\Omega)$ for some $\beta>1$. In addition, we consider the case $q=0$, where the nonlinear boundary condition becomes an elliptic inclusion. Our approach allows us to show that these problems have at least six nontrivial solutions, three positive and three negative, for some positive parameters $\lambda$ and $\mu$. The proof is based on variational arguments.

We study the nonlocal parabolic equation $$ u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s) \, ds $$ with $p>1$. We assume that $k$ is continuous and there exists $\gamma \in {\mathbb{R}}$ such that $k(\lambda t, \lambda s)=\lambda^{-\gamma}k(t,s)$ for all $\lambda>0$, $0<s<t$. We consider the problem in ${\mathbb{R}}^N$ and a Dirichlet problem in a bounded smooth domain $\Omega$. We analyze the conditions for either blow up or global existence of solutions.

We study the global existence of solutions to nonlinear dispersive wave equations \begin{equation*} \partial _{t}^{2}u+\frac{1}{\rho ^{2}}\left\vert \partial _{x}\right\vert ^{2\rho }u=\lambda \left\vert \partial _{t}u\right\vert ^{p-1}\partial _{t}u \end{equation*} in one space dimension, where $0<\rho \leq 2,\rho \neq 1,p>3$ and $\lambda \in \mathbf{C}.$

We consider the Cauchy problem for the Chern-Simons-Dirac system on ${\mathbb{R}}^{1+1}$ with initial data in $H^s$. Almost optimal local well posedness is obtained. Moreover, we show that the solution is global in time, provided that initial data for the spinor component has finite charge, or $L^2$ norm.

We study the oscillation properties of solutions to the nonlinear scalar second-order ODE \begin{equation} u''(t)+\vert u(t) \vert ^{\beta}u(t)+g(u'(t))=0,\quad t\leq0, \end{equation} where $\beta$ is a positive constant and $g:\mathbf{R}\rightarrow \mathbf{R}$ is an increasing and locally Lipschitz function behaving globally like $\vert v \vert^{\alpha}v$, $\alpha>0.$

We consider the Nernst-Planck-type drift-diffusion equation with fractional dissipation. For the initial-value problem of this equation, the well-posedness, the time-global existence, and the decay of solutions were already shown. When the dissipation operator is given by the Laplacian, the asymptotic expansion of the solution as $t\to\infty$ was obtained in a previous paper. We also derive the asymptotic expansion of the solution to the drift-diffusion equation with the fractional Laplacian.

We consider the blow-up problem for a superlinear heat equation $$ \begin{cases} \partial_t u=\epsilon\Delta u+f(u), &x\in\Omega, \,\,\, t>0, \\ u(x,t)=0, &x\in\partial\Omega, \,\,\, t>0 \quad\mbox{if}\quad \partial\Omega\not=\emptyset, \\ u(x,0)=\varphi_\epsilon(x)\ge 0\, (\not\equiv 0), &x\in\Omega, \end{cases} $$ where $\epsilon>0$, $N\ge 1$, $\Omega$ is a domain in ${\bf R}^N$, $f=f(s)$ is a convex function in $s\in (0,\infty)$, and the initial function $\varphi_\epsilon$ is a nonnegative bounded continuous function in $\overline{\Omega}$. The typical examples of $f$ that we treat in this paper, are $f(u)=(u+\lambda)^p$ ($p>1$, $\lambda\ge 0$) and $f(u)=e^u$. In this paper, under suitable assumptions, we prove that the solution $u_\epsilon$ blows up only near the maximum points of the initial function $\varphi_\epsilon$ if $\epsilon>0$ is sufficiently small.

In this paper the method of potential well is used to characterize a set of initial data which do not yield global solutions for a family of nonlocal or higher-order parabolic problems described in (1) and (2).