Fast and slow decay solutions for supercritical elliptic problems in exterior domains

Abstract

We consider the elliptic problem Delta u + u(p) = 0, u > 0 in an exterior domain, Omega = R-N\D under zero Dirichlet and vanishing conditions, where D is smooth and bounded in R-N, N >= 3, and p is supercritical, namely p > N+2/N-2. We prove that this problem has infinitely many solutions with slow decay O(vertical bar x vertical bar(-2/p-1)) at infinity. In addition, a solution with fast decay O(vertical bar x vertical bar(2- N)) exists if p is close enough from above to the critical exponent

Bubbling along boundary geodesics near the second critical exponent

“…Theorem 1.1 includes the case of an exterior domain, \ , with bounded. It is worth mentioning that for this case it was established in [8,9] that problem (1.1) is actually always solvable if p > (n + 2)/(n − 2). In fact a continuum of solutions exist but they are of slow decay (infinite energy).…”

“…In fact a continuum of solutions exist but they are of slow decay (infinite energy). Finding finite-energy (fast decay) solutions for supercritical powers is a much harder question, which is only answered in [9] for p very close from above to (n + 2)/(n − 2). In turns out that a dramatic change of structure in the set of slow decay solutions takes place precisely when p = (n + 1)/(n − 3), the second critical exponent.…”

“…where α and β are positive constants depending only on the dimension N, andh 00 is the normal curvature along the geodesic , which is assumed to be smooth and strictly positive (see For now, we assume they are smooth functions of x 0 and that they have the following norms bounded: 9) where…”

A note on a result of M. Grossi

“…The only fact that needs to be checked is that the function F p defined in Theorem 2 is the same function as (8). Indeed, it is well known that Green's function of the operator…”

“…For supercritical exponents, besides the early nonexistence results, there are few positive (existence) results (see e.g. [8] and the references therein).…”