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EnglishWe 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 $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.}
| Titolo: | Saddle type solutions to a class of semilinear elliptic equations |
| Autori: | |
| Data di pubblicazione: | 2007 |
| Rivista: | |
| 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 $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.} |
| Handle: | http://hdl.handle.net/11566/34140 |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
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