A supercritical elliptic equation in the annulus

Abstract

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equationHere p > 2 is allowed to be supercritical and a(x) is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution u we construct. In the case where a equals a positive constant, we obtain nonradial solutions in the case where the exponent p is large or when the annulus A is large with fixed width.

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

“…We note the results in Corollary 1.3 recovers and complements some of the existing results on the subject (see [35,11] and the references therein).…”

“…Here we adapt the linear theory (Proposition 3.2) and the improved Sobolev imbedding on K from [11] to our setting. To be specific define the convex set K by…”

“…We now write the equations in the variables (r, θ) (for details regarding these variables see [11]). Using the facts…”

“…In the purely critical problem p = 2 * it was shown in [2] that there is a solution provided the domain is not contractible. However, finding positive non-radial solution for the supercrital case has been a very delicate issue as well; for instance we refer to the recent paper [11]. Before providing a more comprehensive background on this problem, we shall begin with our contribution on the subject in this paper.…”

Asymptotics for a high-energy solution of a supercritical problem

“…Recently it has been proved that, even restricting one's attention to the search for radial solutions, this problem presents multiplicity and the structure of the set of radial solutions can be very rich, depending on the values of the parameters into play. On the other hand, very little is still known in the non-radial setting; for p = 2 we refer to [14,15] for the existence of solutions in non-radial domains, and to [9] for the existence of a non-radial solution of the Dirichlet problem in an annulus.…”