The symmetry of least-energy solutions for semilinear elliptic equations. (English) Zbl 1247.35038
Summary: 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:
\[
\Delta u+K(x)u^p=0, \quad u>0\text{ in }B,\quad u=0\text{ on }\partial B, \tag{*}
\]
where \(1<p<(n+2)/(n-2)\) and \(B\) is the unit ball in \(\mathbb R^n\) with \(n\geq 3\). 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 Z.-C. Han [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 2, 159–174 (1991; Zbl 0729.35014)] to the case when \(K(x)\) is nonconstant. In contrast to previous works for this kind of estimates, we only assume that \(K(x)\) is continuous.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 47J30 | Variational methods involving nonlinear operators |
Citations:
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