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Nondegeneracy of the saddle solution of the Allen-Cahn equation. (English) Zbl 1241.35079
Authors’ abstract: “A solution of the Allen-Cahn equation in the plane $$\Delta u+f(u)=0$$ (where $$f\in C^2$$, $$f(\pm 1)=f(0)=0$$ and $$f(u)/u$$ is decreasing on $$(0,1)$$) is called a saddle solution if its nodal set coincides with the coordinate axes. Such solutions are known to exist for a large class of nonlinearities. In this paper we consider the linear operator obtained by linearizing the Allen-Cahn equation around the saddle solution: $$Lh\equiv \Delta h+f'(U)h$$, where $$LU_x=LU_y=0$$. Our result states that there are no nontrivial, decaying elements in the kernel of this operator. In other words, the saddle solution of the Allen-Cahn equation is nondegenerate.”

##### MSC:
 35J61 Semilinear elliptic equations 47B34 Kernel operators 35B08 Entire solutions to PDEs
##### Keywords:
Allen-Cahn equation; nondegeneracy
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