Dimension threshold for multi-peak concentration
Determine whether the dimensional restriction n≥9 in the construction of multi-peak solutions for the zero-mass singularly perturbed problem -ε²Δu=u^p−Q(x)u^q is merely technical, or whether a different phenomenon occurs in lower dimensions.
References
Several natural questions remain open. First, our argument requires the dimensional restriction n\geq9. This condition enters through the estimates of the algebraic interactions and of the error terms, and it would be interesting to understand whether it is merely technical or whether a different phenomenon occurs in lower dimensions.
Several natural questions remain open. First, our argument requires the dimensional restriction n\geq9. This condition enters through the estimates of the algebraic interactions and of the error terms, and it would be interesting to understand whether it is merely technical or whether a different phenomenon occurs in lower dimensions. Second, the present construction requires q to lie above the threshold q_1>n/(n-2). It is natural to ask whether concentration near arbitrary nondegenerate critical points persists throughout the larger range \frac{n}{n-2}<q<p. Finally, in the classical positive-mass problem, semiclassical concentration can be obtained under substantially weaker geometric assumptions than nondegeneracy of an isolated critical point. It would therefore be interesting to investigate the zero-mass problem in the presence of degenerate critical points or critical manifolds of Q.