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Deformation Lemma, Ljusternik-Schnirellmann Theory and Mountain Pass Theorem on C1-Finsler Manifolds

Ribarska, Nadezhda, Tsachev, Tsvetomir, Krastanov, Mikhail (1995)

Serdica Mathematical Journal

∗Partially supported by Grant MM409/94 Of the Ministy of Science and Education, Bulgaria. ∗∗Partially supported by Grant MM442/94 of the Ministy of Science and Education, Bulgaria.Let M be a complete C1−Finsler manifold without boundary and f : M → R be a locally Lipschitz function. The classical proof of the well known deformation lemma can not be extended in this case because integral lines may not exist. In this paper we establish existence of deformations generalizing the classical result. This...

Editorials' note

Miroslav Fiedler, Pankaj Jain, Lars-Erik Persson (2009)

Czechoslovak Mathematical Journal

Eigenvalues of the p -Laplacian in 𝐑 N with indefinite weight

Yin Xi Huang (1995)

Commentationes Mathematicae Universitatis Carolinae

We consider the nonlinear eigenvalue problem - div ( | u | p - 2 u ) = λ g ( x ) | u | p - 2 u in 𝐑 N with p > 1 . A condition on indefinite weight function g is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in W 1 , p ( 𝐑 N ) . A nonexistence result is also given for the case p N .

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems ( P ε ) : Δ u ε = u ε ( n + 2 ) / ( n 2 ) , u ε > 0 in A ε , u ε = 0 on A ε , where A ε is an annulus-shaped domain of n , n 3 , which becomes thinner as ε 0 . We show that for every solution u ε , the energy A ε | u | 2 as well as the Morse index tend to infinity as ε 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on n , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

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