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 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.