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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
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