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Advances in Calculus of Variations

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Volume 11, Issue 3


𝒟1,2(ℝN) versus C(ℝN) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝN

Siegfried Carl / David G. Costa
  • Corresponding author
  • Department of Mathematical Sciences, University of Nevada Las Vegas, Box 454020, Las Vegas, USA
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/ Hossein Tehrani
Published Online: 2017-04-08 | DOI: https://doi.org/10.1515/acv-2016-0012


We consider functionals of the form


on a C1-submanifold M of 𝒟1,2(N), N3, where G is the primitive of some “zero-mass” nonlinearity g (i.e., g(0)=0), and the weight function b:N is merely supposed to belong to L1(N)L2*2*-p(N) for some 2<p<2*, and to possess a certain decay behavior. Let V be the subspace of 𝒟1,2(N) given by V:={v𝒟1,2(N):vC(N) with supxN(1+|x|N-2)|v(x)|<}. We prove that a local minimizer of the constrained functional J|M with respect to the V-topology must be a local minimizer with respect to the “bigger” 𝒟1,2(N)-topology. This result allows us to prove the existence of multiple nontrivial solutions of the zero-mass equation -Δu=b(x)g(u) in N, where g:R is a subcritical nonlinearity, which is superlinear at zero and at .

Keywords: Minimizer on manifold; multiple solutions; zero-mass equations

MSC 2010: 35J15; 35J20


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About the article

Received: 2016-03-20

Accepted: 2017-03-04

Published Online: 2017-04-08

Published in Print: 2018-07-01

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 3, Pages 257–272, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2016-0012.

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