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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
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