Connecting orbits for some classes of almost periodic Lagrangian systems

The existence is proved, by means of variational arguments, of infinitely many heteroclinic solutions connecting possibly degenerate equilibria for a class of almost periodic Lagrangian system. An analogous multiplicity result is then established for homoclinic solutions of systems with an almost periodic singular potential.ou

An energy constrained method for the existence of layered type solutions of NLS equations

We study the existence of positive solutions on $\R^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f(u)$ where $N\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of \f(u)=\tfrac 12\|u\|^{2}_{H^{1}(\R^{N})}-\int_{\R^{N}}F(u)\, dx, $u\in H^{1}(\R^{N})$ ($F(s)=\int_{0}^{s}f(t)\, dt$), we show that for any $b\in [0,c)$ there exists a positive bounded solution $v_{b}\in C^{2}(\R^{N+1})$ such that $E_{v_{b}}(y)=\tfrac 12\|\partial_{y}v_{b}(\cdot,y)\|^{2}_{L^{2}(\R^{N})}-V(v_{b}(\cdot,y))=-b$. We also characterize the monotonicity, symmetry and periodicity properties of $v_{b}$

Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations

We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+a(\varepsilon x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where \e>0, $a:\R\to\R$ is an almost periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show via variational methods that if \e is sufficiently small and $a$ is not constant then the equation admits infinitely many two dimensional entire solutions verifying the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$