A generalized mountain-pass theorem: the existence of multiple critical points

We analyse the existence of multiple critical points for an even functional J : H -> R in the following context: the Hilbert space H can be split into an orthogonal sum H = Y circle plus Z in such a way that inf{J(u) : u is an element of Z and parallel to u parallel to = rho >= alpha > J(0) and that here exits a point b is an element of H with parallel to b parallel to > rho and with J(b) <= J(0). We develop a new variational characterization of multiple critical levels without an assumption on the dimension of Y. Our characterization is simple and natural: we can for example avoid the notion of pseudo-index and the definition of the activated levels does not substantially differs from the one used for the lowest critical level, giving us in this way a unified view of critical levels. We apply our results to a semi-linear Schrodinger equation of the form {-Delta u + V(x)u - q(x)vertical bar u vertical bar(sigma)u = lambda u, x is an element of R-N u is an element of H-1(R-N)\{0} where lambda is inside a spectral gap bounded on both sides by parts of the essential spectrum.

Published in:
Calculus Of Variations And Partial Differential Equations, 55, 4, 89
Heidelberg, Springer Heidelberg

 Record created 2016-10-18, last modified 2018-01-28

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