## The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent.(English)Zbl 0786.35059

Let $$\Omega$$ be a smooth and bounded domain in $$\mathbb{R}^ N$$, $$N \geq 4$$, and $$p=(N+2)/(N-2)$$ so that $$p+1$$ is critical from the viewpoint of Sobolev embedding. The authors consider the nonlinear elliptic problems of the type $$(P_ \varepsilon):-\Delta u=u^{(N+2)/(N-2)}+\varepsilon u$$, $$u>0$$ on $$\Omega$$; $$u=0$$ on $$\partial \Omega$$, and $$\varepsilon>0$$. They show that if the $$u_ \varepsilon$$ are solutions of $$(P_ \varepsilon)$$ which concentrate around a point as $$\varepsilon \to 0$$, then this point cannot be on the boundary of $$\Omega$$ and is a critical point of the regular part of the Green’s function. Conversely, they show that for $$N \geq 5$$ and any nondegenerate critical point $$x_ 0$$ of the regular part of the Green’s function, there exist solutions of $$(P_ \varepsilon)$$ concentrating around $$x_ 0$$ as $$\varepsilon \to 0$$.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B25 Singular perturbations in context of PDEs 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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### References:

 [1] Alexander, J.; Yorke, J., The homotopy continuation method: Numerically implémentable topological procedures, Trans. Amer. Math. Soc., 242, 271-284 (1978) · Zbl 0424.58003 [2] Aubin, Th, Nonlinear Analysis on Manifolds, Monge-Ampère Equations, (Grundlheren, Vol. 252 (1982), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0512.53044 [4] Bahri, A.; Coron, J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41, 253-290 (1988) · Zbl 0649.35033 [5] Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d’une variété riemannienne, (Lecture Note in Mathematics, Vol. 194 (1971), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0141.38203 [6] Brezis, H., Some variational problems with lack of compactness, (Proc. Sympos. Pure Math., 45 (1986)), 165-201 [7] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029 [9] Donaldson, S., Connections, cohomology and the intersection forms of 4-manifolds, J. Differential Geom., 24, 275-341 (1986) · Zbl 0635.57007 [10] das, B. G.; Ni, W.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020 [12] Rabinowitz, P., Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3, 161-202 (1973) · Zbl 0255.47069 [13] Schoen, R., Conformal deformations of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20, 479-495 (1984) · Zbl 0576.53028 [14] Struwe, M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187, 511-517 (1984) · Zbl 0535.35025 [15] Taubes, C., Self-dual connections on manifolds with indefinite intersection matrix, J. Differential Geom., 19, 517-560 (1984) · Zbl 0552.53011
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