1,720,992 research outputs found
Periodic and heteroclinic type solutions for systems of Allen-Cahn equations
No full textWe consider a class of semilinear elliptic system of the form on , where is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system which connect the two minima of as has a discrete structure, then the two dimensional system has infinitely many layered solutions with prescribed energy
Stationary layered solutions for a system of Allen-Cahn type equations
No full textWe consider a class of semilinear elliptic system of the form
−∆u(x,y)+∇W(u(x,y))=0, (x,y)∈R2,
where W : R2 → R is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system −q ̈(x) + ∇W(q(x)) = 0, x ∈ R, which connect the two minima of W as x → ±∞ has a discrete structure, then the system has infinitely many layered solutions
Homoclinic solutions for second order systems with expansive time dependence
No full textSi dimostra l'esistenza di almeno una soluzione omoclina per sistemi
Lagrangiani della forma in
dove
è superquadratica e
soddisfa la condizione .
Il metodo è variazionale: le soluzioni omocline del sistema risultano
essere punti critici di un opportuno funzionale d'azione. Si dimostra
l'esistenza di almeno un punto critico non banale usando l'analisi
dei pmblemi \textquotedbl{}all'infinito\textquotedbl{} e argomenti
di confronto sui livelli.We prove the existence of homoclinic solutions for second order Lagrangian
systems of the type
in where
is superquadratic and
satisfies the condition .
The method is variational solutions being found as critical points
of a suitable action functional. We prove the existence of al least
one non-trivial critical point using the analysis of problems \textquotedbl{}at
infinity\textquotedbl{} and level comparison arguments
Brake orbits type solutions to some class of semilinear elliptic equations
No full textWe consider a class of
semilinear elliptic equations of the form
where is a periodic, positive function and
is modeled on the classical two well Ginzburg-Landau
potential . We show, via variational methods, that
if the set of solutions to the one dimensional heteroclinic problem
has a discrete structure, then
the equation has infinitely many solutions periodic in the variable
and verifying the
asymptotic conditions as
uniformly with respect to
Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in R^{3}
No full textWe consider a class of semilinear elliptic equations of the form
\begin{equation}\label{eq:abs}
-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\R^{3},
\end{equation}
where is a periodic, positive, even function and, in the simplest case, is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the associated one dimensional heteroclinic problem we show, via variational methods the existence of infinitely many geometrically distinct solutions of (\ref{eq:abs}) verifying as uniformly with respect to and such that , in
Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential
No full textWe consider a class of semilinear elliptic system of the form -Delta u(x,y)+
abla W(u(x,y))=0,quad (x,y)inR^{2},
where W:R^{2} oR is a double well potential with minima a_pminR^$. We show, via variational methods, that
if the set of minimal heteroclinic solutions to the one dimensional system -ddot q(x)+
abla W(q(x))=0, xinR, up to translations, is finite and constituted by not degenerate functions, then the system has infinitely many solutions uin C^{2}(R^{2})^{2}, parametrized by an energy value, which are periodic in the variable y and satisfy lim_{x opminfty}u(x,y)=a_{pm} for any yinR
Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations
No full textWe consider a class of
semilinear elliptic equations of the form
where \e>0, is an almost periodic, positive function and
is modeled on the classical two well Ginzburg-Landau
potential . We show via variational
methods that if \e is sufficiently small and is not constant
then the equation admits infinitely many two dimensional entire solutions
verifying the asymptotic conditions as
uniformly with respect to
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