On nodal radial solutions of an elliptic problem involving critical Sobolev exponent. (English) Zbl 0853.35033

The main result of this paper is the following theorem: let \(K\) and \(f\) be continuous functions on \([0, \infty)\) and \(\mathbb{R}\), resp., satisfying \(K(0)= \lim_{|x|\to \infty} K(|x|)= 0\), \(K(|x|)> 0\) on \(\mathbb{R}^n\backslash \{0\}\), and \(f(u) u\geq \mu \int^u_0 f(s) ds\), \(|f(u)|\leq a|u|^{2^*- 1}\) for some constants \(2< \mu< 2^*:= 2n/(n- 2)\), \(a> 0\), and all \(u\in \mathbb{R}\). Assume that \(f(u)/|u|\) is strictly increasing on \(\mathbb{R}\backslash \{0\}\). Then for every integer \(j\geq 0\) the equation \(- \Delta u= K(|x|) f(u)\) in \(\mathbb{R}^n\), \(n> 2\), admits radial solutions \(u^-_j\), \(u^+_j\) with \(u^-_j(0)< 0< u^+_j(0)\), having exactly \(j\) nodes.
The first step in the proof is to show the existence of positive and negative solutions \(u^+\) and \(u^-\) of the Dirichlet problem \(-\Delta u= K(|x|) f(u)\) in \(Q\), \(u(x)= 0\) on \(\partial Q\), where \(Q\) is either a ball, an annulus, or the exterior of a ball. \(u^+\) and \(u^-\) are obtained as critical points of appropriate variational functionals \({\mathcal F}^+\) and \({\mathcal F}^-\) by the mountain pass theorem of A. Ambrosetti and P. H. Rabinowitz [J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)]. To prove now the existence of a radial solution having a prescribed number of nodes, the author needs the additional information \({\mathcal F}^{\pm}(u^{\pm})= \inf_{u\in {\mathcal N}^{\pm}} {\mathcal F}^{\pm}(u)\), where \({\mathcal N}^{\pm}\) is the Nehari manifold associated with \({\mathcal F}^{\pm}\). This statement follows from a more general result about potential operator equations in reflexive Banach spaces, which is proved in the beginning of the paper.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations


Zbl 0273.49063
Full Text: EuDML