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: Δu+K(x)u ^P = 0, XEB _1, u>0 in b _1, u _∂B1 = 0, where 1 < p < n+2/n-2 and B ₁ is the unit ball of ℝ ⁿ 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 n+2/n-2, 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.