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Perturbation of \(\Delta u+u^{(N+2)/(N-2)}=0\), the scalar curvature problem in \(\mathbb{R}^N\), and related topics. (English) Zbl 0938.35056

Let \(p=(n+2)/(n-2)\) be the critical Sobolev exponent in \(\mathbb{R}^n\), \(n\geq 3\). This paper deals with equations of the form \[ -\Delta u=u^p+ \varepsilon F(x,u)\text{ on }\mathbb{R}^n,\tag{1} \] where \(F(x,u)\) is alternately equal to \(K(x)u^p\) (resp. \(K(r)u^p,r=|x|\), i.e. \(K\) is radial), \(K(x)u^p+h(x)u\), with \(n>4\) and \(h(x)u^q\) with \(1<q<p\), \(n\geq 3\). In each case, the authors study conditions on the data which imply the existence of at least one positive solution of (1), in a suitable space, when \(\varepsilon\) is small enough. Some results are related with the scalar curvature problem in \(\mathbb{R}^n\). Their approach is based on the abstract perturbation method in critical point theory discussed in previous papers by the first author and M. Badiale [Ann. Inst. H. Poincaré, Anal. Non Linéaire 15, 233-252 (1998) and Proc. R. Soc. Edinb., Sect. A, Math. 128 No, 6, 1131-1161 (1998; Zbl 0928.34029)].
Reviewer: D.Huet (Nancy)

MSC:

35J60 Nonlinear elliptic equations
35B20 Perturbations in context of PDEs
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