The symmetry of least-energy solutions for semilinear elliptic equations

In this paper we will apply the method of rotating planes (MRP) to investigate the radial and axial symmetry of the least-energy solutions for semilinear elliptic equations on the Dirichlet and Neumann problems, respectively. MRP is a variant of the famous method of moving planes. One of our main results is to consider the least-energy solutions of the following equation: where and B1 is the unit ball of with n 3. Here K(x)=K(x) is not assumed to be decreasing in x. In this paper, we prove that any least-energy solution of (*) is axially symmetric with respect to some direction. Furthermore, when p is close to , under some reasonable condition of K, radial symmetry is shown for least-energy solutions. This is the example of the general phenomenon of the symmetry induced by point-condensation. A fine estimate for least-energy solution is required for the proof of symmetry of solutions. This estimate generalizes the result of Han (Ann. Inst. H. Poincaré Anal. Nonlinéaire 8 (1991) 159) to the case when K(x) is nonconstant. In contrast to previous works for this kinds of estimates, we only assume that K(x) is continuous.