## 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

### Citations:

Zbl 0980.21982; Zbl 0928.34029
Full Text:

### References:

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