Slope-changing solutions of elliptic problems on image Original Research Article

We consider the problem View the MathML source−Δu+Fu(x,u)=0 on View the MathML sourceRn, where FF is a smooth function periodic of period 1 in all its variables. We are going to find a non-degeneracy condition on FF for which the following holds. If we are given a sequence of positive integers View the MathML source{Ñi}i∈Z and a sequence View the MathML source{αi}i∈Z of real numbers (the slopes), then we shall find an increasing sequence {Qi}{Qi} of integers and a solution uu which is entire, periodic in (x2,…,xn)(x2,…,xn) and which is close to the plane α1(x1−Qi)+u(Qi,0,…,0)α1(x1−Qi)+u(Qi,0,…,0) for View the MathML sourcex1∈[Qi,Qi+Ñi].