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We prove the existence of infinitely many geometrically distinct homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems.

We consider a class of semilinear elliptic equations of the form $$-{\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$$ where $\epsilon \>0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to (1) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show via variational methods that if $\epsilon$ is sufficiently small and $a$ is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one...

We consider a class of semilinear elliptic equations of the form 15.7cm -${\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$ where $\epsilon >0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show variational methods that if is sufficiently small and is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct up...

We show, by variational methods, that there exists a set $\mathcal{A}$ open and dense in $a\in L^{\mathrm{\infty }}\left({\mathbb{R}}^N\right):a\ge 0$ such that if $a\in \mathcal{A}$ then the problem $-\mathrm{△}u+u=a\left(x\right){\left|u\right|}^{p-1}u,u\in H^1\left({\mathcal{R}}^N\right)$, with $p$ subcritical (or more general nonlinearities), admits infinitely many solutions.

Si considera una classe di equazioni ellittiche semilineari su ${\mathbb{R}}^N$ della forma $-\mathrm{\Delta }u+u=a\left(x\right){\left|u\right|}^{p-1}u$ con $p>1$ sottocritico (o con nonlinearità più generali) e $a\left(x\right)$ funzione limitata. In questo articolo viene presentato un risultato di genericità sull'esistenza di infinite soluzioni, rispetto alla classe di coefficienti $a\left(x\right)$ limitati su ${\mathbb{R}}^N$ e non negativi all'infinito.