## 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.

### MSC:

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

Zbl 0273.49063
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