We consider a class of semilinear elliptic equations of the form $$ -\Delta u(x,y)+W'(u(x,y))=0,\quad (x,y)\in\R^{2} $$ where $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show, via variational methods, that for any $j\geq 2$, the equation has a solution $v_{j}\in C^{2}(\R^{2})$ with $|v_{j}(x,y)|\leq 1$ for any $(x,y)\in\R^{2}$ satisfying the following symmetric and asymptotic conditions: setting $\tilde v_{j}(\rho,\theta)=v_{j}(\rho\cos( \theta),\rho\sin(\theta))$, there results $\tilde v_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde v_{j}(\rho,\frac{\pi}{2}-\theta)$, $\tilde v_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde v_{j}(\rho,\theta)$, $\forall (\rho,\theta)\in \R^{+}\times\R$ and $\tilde v_{j}(\rho,\theta)\to 1$ as $\rho\to+\infty$ for any $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.}

Saddle type solutions to a class of semilinear elliptic equations

ALESSIO, FRANCESCA GEMMA;CALAMAI, Alessandro;MONTECCHIARI, Piero
2007

Abstract

We consider a class of semilinear elliptic equations of the form $$ -\Delta u(x,y)+W'(u(x,y))=0,\quad (x,y)\in\R^{2} $$ where $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show, via variational methods, that for any $j\geq 2$, the equation has a solution $v_{j}\in C^{2}(\R^{2})$ with $|v_{j}(x,y)|\leq 1$ for any $(x,y)\in\R^{2}$ satisfying the following symmetric and asymptotic conditions: setting $\tilde v_{j}(\rho,\theta)=v_{j}(\rho\cos( \theta),\rho\sin(\theta))$, there results $\tilde v_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde v_{j}(\rho,\frac{\pi}{2}-\theta)$, $\tilde v_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde v_{j}(\rho,\theta)$, $\forall (\rho,\theta)\in \R^{+}\times\R$ and $\tilde v_{j}(\rho,\theta)\to 1$ as $\rho\to+\infty$ for any $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.}
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11566/34140
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 31
  • ???jsp.display-item.citation.isi??? ND
social impact