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A global compactness result for elliptic boundary value problems involving limiting nonlinearities. (English) Zbl 0535.35025
For $$n\geq 3$$, $$\Omega \subset {\mathbb{R}}^ n$$ open and bounded, $$\lambda\in {\mathbb{R}}$$ and $$2^*=2n/(n-2)$$ compactness properties of the functional $$E(u)=(1/2)\int_{\Omega}| \nabla u|^ 2-\lambda u^ 2dx- (1/2^*)\int_{\Omega}| u|^ 2dx, u\in H_ 0\!^{1,2}(\Omega)$$ are considered. Relative compactness of sequences $$\{u_ m\}$$ in $$H_ 0\!^{1,2}(\Omega)$$ satisfying $$E(u_ m)\leq c, dE(u_ m)\to 0$$ in $$H^{-1}(\Omega)$$ is shown to depend on the ”spectrum” of energies of solutions to the ”limiting problem” $$-\Delta u=u| u|^{2^*-2}$$ in $${\mathbb{R}}^ n$$ u(x)$$\to 0$$ ($$| x| \to \infty)$$. Moreover, if ”jumps” in the topological type of admissible functions are permitted, compactness of such sequences is established globally.

##### MSC:
 35J60 Nonlinear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
##### Keywords:
global compactness; limiting nonlinearities
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##### References:
 [1] Aubin, T.: Nonlinear analysis on manifolds, Monge-Amp?re equations. Grundlehren252. Berlin Heidelberg New York: Springer 1982 · Zbl 0512.53044 [2] B?hme, R.: Die L?sung der Variationsgleichungen f?r nichtlineare Eigenwertprobleme. Math. Z.123, 105-126 (1972) · Zbl 0254.47082 · doi:10.1007/BF01112603 [3] Brezis, H., Coron, J.-M.: Convergence de solutions deH-syst?mes et applications aux surfaces ? courbure moyenne constante, preprint [4] Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math.36 (1983) · Zbl 0541.35029 [5] Cerami, G., Fortunato, D., Struwe, M.: Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, preprint · Zbl 0568.35039 [6] Donaldson, S.K.: An application of gauge theory to four dimensional topology. J. Diff. Geom.18, 279-315 (1983) · Zbl 0507.57010 [7] Eisen, G.: A selection lemma for measurable sets, and lower semicontinuity of multiple integrals. Manuscripta Math.27, 73-79 (1979) · Zbl 0404.28004 · doi:10.1007/BF01297738 [8] Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys.68, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125 [9] Lions, P.-L.: Applications de la m?thode de concentration-compacit? ? l’existence de fonctions extr?males. C.R. Acad. Sc. Paris296, 645-648 (1983) · Zbl 0522.49008 [10] Marino, A.: La biforcazione nel caso variazionale. Confer. Sem. Nat. Univ. Bari,132 (1973) [11] Palais, R.S.: Morse theory on Hilbert manifolds. Topology2, 299-340 (1963) · Zbl 0122.10702 · doi:10.1016/0040-9383(63)90013-2 [12] Poho?aev, S.I.: Eigenfunctions of the equation ?u+?f(u)=0. Dokldy165, 1408-1412 (1965) [13] Rabinowitz, P.H.: Some aspects of nonlinear eigenvalue problems. Rocky Mount. J. Math.3, 161-202 (1973) · Zbl 0255.47069 · doi:10.1216/RMJ-1973-3-2-161 [14] Sacks, P.-K. Uhlenbeck: On the existence of minimal immersions of 2-spheres. Ann. Math.113, 1-24 (1981) · Zbl 0462.58014 · doi:10.2307/1971131 [15] Struwe, M.: LargeH-surfaces via the mountain-pass-lemma. preprint · Zbl 0582.58010 [16] Wente, H.C.: Large solutions to the volume constrained Plateau problem. Arch. Rational Mech. Anal.75, 59-77 (1980) · Zbl 0473.49029 · doi:10.1007/BF00284621
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