Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains

In this paper, we provide an affirmative answer to the conjecture A for bounded simple rotationally symmetric domains Ω⊂ Rⁿ(n≥ 3) along xₙ axis. Precisely, we use a new simple argument to study the symmetry of positive solutions for two kinds of semilinear elliptic equations. To do this, when f(·,s) is strictly convex with respect to s, we show that the nonnegativity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of u. Moreover, we prove the uniqueness of critical points of a positive solution to semilinear elliptic equation - u=f(·,u) with zero Dirichlet boundary condition for simple rotationally symmetric domains in Rⁿ by continuity method and a variety of maximum principles.