Multi-Peak Solutions for Super-Critical Elliptic Problems in Domains with Small Holes

Pino, Manuel del; Felmer, Patricio; Musso, Monica

doi:10.1006/jdeq.2001.4098

Cited by 54 publications

(7 citation statements)

References 15 publications

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“…Except for results in domains involving symmetries or exponents close to critical, see for instance [7,8,10,14,16], solvability of (1.1)- (1.2) in the supercritical case has been a widely open matter, particularly since variational machinery no longer applies, at least in its naturally adapted way for subcritical or critical problems.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 98%

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

et al. 2008

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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

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 98%

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

et al. 2008

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“…To our knowledge the only other result in this direction has been obtained in [7] for symmetric domains. In the supercritical case Theorem 1.1 is the first existence result for a positive solution without assuming that has a small hole as in [11,12].…”

Section: Remark 12mentioning

confidence: 99%

Asymptotically radial solutions in expanding annular domains

et al. 2011

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In this paper we consider the problemwhere p > 1 and R is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as R −→ ∞. The main goal of the paper is to prove, for large R, the existence of a positive solution to (0.1) which is close to the positive radial solution in the corresponding diffeomorphic annulus. The proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions.M. Grossi and F. Pacella were supported by M.I.U.R., project "Variational methods and nonlinear differential equations". M.

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“…(1.7) has no non-trivial solution if p(r ) = 2 * + c, with c ≥ 0, while we obtain the existence of a solution for any p(r ) of the form p(r ) = 2 * + r α . For existence results concerning equations with critical growth and with conditions on the domain, we refer to the results of Bahri-Coron [1] and Coron [3], while for equations with slightly supercritical growth we recall the articles of del Pino [4], del Pino-Wei [7], see also [5,6], and for other types of solutions in supercritical equations [11,12].…”

Section: Letmentioning

confidence: 99%

On supercritical Sobolev type inequalities and related elliptic equations

Ruf

Ubilla

2016

Calc. Var.

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Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. In particular, the following inequality is proved: let B ⊂ R N , N ≥ 3, be the unit ball, and let H 1 0,rad (B) denote the first order Sobolev space of radial functions, and 2 * = 2N /(N − 2) the corresponding critical Sobolev embeddding exponent. Let r = |x|, and p(r ) = 2 * + r α , with α > 0; thenWe point out that the growth of p(r ) is strictly larger than 2 * , except in the origin. Furthermore, we show that for p(r ) = 2 * + r α , with 0 < α < min{ N 2 ; N − 2}, the supremum in (0.1) is attained. Finally, we prove that associated elliptic equations admit nontrivial radial solutions. This is somewhat surprising since the nonlinearities have strictly supercritical growth except in the origin. Mathematics Subject Classification35J20 · 35J25 · 35J50 Communicated by A. Malchiodi. Research supported in part by INCTmat/MCT/Brazil,

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Multi-Peak Solutions for Super-Critical Elliptic Problems in Domains with Small Holes

Cited by 54 publications

References 15 publications

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

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

Asymptotically radial solutions in expanding annular domains

On supercritical Sobolev type inequalities and related elliptic equations

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