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In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msup><mml:mrow><mml:mi>\u211d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math>. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.<\/jats:p>","DOI":"10.1155\/2012\/102489","type":"journal-article","created":{"date-parts":[[2012,11,13]],"date-time":"2012-11-13T21:00:22Z","timestamp":1352840422000},"page":"1-32","source":"Crossref","is-referenced-by-count":2,"title":["Variational Methods for NLEV Approximation Near a Bifurcation Point"],"prefix":"10.1155","volume":"2012","author":[{"given":"Raffaele","family":"Chiappinelli","sequence":"first","affiliation":[{"name":"Dipartimento di Scienze Matematiche ed Informatiche, Universit\u00e0 di Siena, Pian dei Mantellini 44, 53100 Siena, Italy"}]}],"member":"98","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141096302111"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1080\/03605308708820534"},{"key":"34a","first-page":"133","volume":"15","year":"2010","journal-title":"Ulmer Seminar"},{"key":"30","first-page":"xxi+619","year":"1976"},{"key":"50","first-page":"v+160","year":"1974"},{"key":"31","series-title":"Translated by A. 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