zbMATH — the first resource for mathematics

Saddle-type solutions for a class of semilinear elliptic equations. (English) Zbl 1193.35058
The class of semilinear elliptic equations $-\Delta u(x,y)+W'(u(x,y))=0,\quad (x,y)\in{\mathbb R}^{2}$ is considered in the case in which $$W:{\mathbb R}\to{\mathbb R}$$ is a bistable symmetric potential. Variational methods are used to reveal the existence of infinitely many saddle-type entire solutions. Precisely it is shown that for any $$j\geq 2$$, the equation has a solution $$v_{j}$$ on $${\mathbb R}^{2}$$ satisfying the following symmetric and asymptotic conditions: setting $$\tilde v_{j}(\rho,\theta)=v_{j}(\rho\cos( \theta),\rho\sin(\theta))$$, there results
i) $$\tilde v_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde v_{j}(\rho,\frac{\pi}{2}-\theta)$$ and
ii) $$\tilde v_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde v_{j}(\rho,\theta)$$, $$\forall (\rho,\theta)\in {\mathbb R}^{+}\times{\mathbb R}$$ and $$\tilde v_{j}(\rho,\theta)\to 1$$ as $$\rho\to+\infty$$ for any $$\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$$.

MSC:
 35J61 Semilinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35J20 Variational methods for second-order elliptic equations