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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 \(\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)\).

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
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