Nonlinear Elliptic Equations on Expanding Symmetric Domains

Abstract

In this article we study the problemin the case 0 a is an expanding domain. In particular, for n 2 when 0 a = [x # R n : a< |x| <a+1] is an expanding annulus as a Ä , we prove the existence of many rotationally non-equivalent solutions obtained as local minimizers of the corresponding energy functional. Moreover, we study the exact symmetry and the shape of these solutions, and under certain conditions we prove the existence of solutions with prescribed symmetry. Academic Press

Asymptotically radial solutions in expanding annular domains

“…In the case of a nonlinearity with subcritical growth our theorem provides a new multiplicity result since, besides the positive multibump solutions constructed in [1,5,10], it asserts the existence of asymptotically radial solutions in a different type of expanding domains. In the critical exponent case our approach gives a direct proof of the Bahri-Coron result [3] for annular shaped domains with large holes, complementing various results in domains with small holes [6,9,17,24].…”

“…In the special case when is an annulus it is easy to prove that a radial positive solution always exists, whatever p is, even supercritical (see [16]), and this solution is unique (see [21]). Moreover, exploiting the invariance of the annulus with respect to different symmetry groups, several authors were able to prove the existence of nonradial positive solutions for p up to a certain exponent p N > N +2 N −2 in expanding annuli A R = x ∈ R N : R < |x| < R + 1 , for R sufficiently large (see [4,5,8,[18][19][20]). A study of the asymptotic behavior of some of these solutions, as R → ∞, shows that they converge to positive solutions on an infinite strip (see [20]).…”

“…in domains with an expanding hole, which are not necessarily annuli. This idea has been carried out in the papers [1,5,10] where, in the subcritical case, the existence of an increasing number of positive solutions is proved, as the domain expands. In particular, in [10] the limit problem in the strip is exploited to construct positive multibump solutions using a LyapunovSchmidt reduction argument.…”

“…So they are diffeomorphic to an annulus. However the solutions that they construct (as well as those of [5]) are very different from radial solutions in annuli, since they exhibit a finite number of bumps.…”

Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus

“…Let us recall that previous results on the existence of nonradial solutions of (1.1) in expanding annuli were obtained in [2,3,9] in the case when p is a subcritical exponent, 1 < p < N +2 N −2 , N ≥ 3 or for a small range of supercritical exponent. Note that for annuli with small holes it has been proved in [7] that, in the subcritical case, all positive solutions are radial.…”