Uniqueness of positive periodic solutions with some peaks

الأصل · EN

This work deals with the semi linear equation -Δu+u-uᵖ=0 in ⁿ, 2≤ p<N+2 N-2. We consider the positive solutions which are 2π-periodic in x₁ and decreasing to 0 in the other variables, uniformly in x₁. Let a periodic configuration of points be given on the x₁-axis, which repel each other as the period tends to infinity. If there exists a solution which has these points as peaks, we prove that the points must be asymptotically uniformly distributed on the x₁-axis. Then, for small enough, we prove the uniqueness up to a translation of the positive solution with some peaks on the x₁-axis, for a given minimal period in x₁.