Abstract
In this paper we prove existence and multiplicity of positive and sign-changing solutions to the pure critical exponent problem for the \(p\)-Laplacian operator with Dirichlet boundary conditions on a bounded domain having nontrivial topology and discrete symmetry. Pioneering works related to the case \(p=2\) are Brezis and Nirenberg (Comm Pure Appl Math 36, 437–477, 1983), Coron (C R Acad Sci Paris Sr I Math 299, 209–212, 1984), and Bahri and Coron (Comm. Pure Appl. Math. 41, 253–294, 1988). A global compactness analysis is given for the Palais-Smale sequences in the presence of symmetries.
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C. M. would like to thank the Department of Mathematics “Guido Castelnuovo” (Universitá di Roma “Sapienza”) for the kind hospitality when this paper has been started.
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Communicated by A. Malchiodi.
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Mercuri, C., Pacella, F. On the pure critical exponent problem for the \(p\)-Laplacian. Calc. Var. 49, 1075–1090 (2014). https://doi.org/10.1007/s00526-013-0612-x
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DOI: https://doi.org/10.1007/s00526-013-0612-x