Ljusternik–Schnirelmann Theorem for the Generalized Laplacian

AbstractLet m:[0, ∞[→[0, ∞[ be an increasing continuous function with m(t)=0 if and only if t=0, m(t)→∞ as t→∞ and Ω⊂RN a bounded domain. In this paper we show that for every r>0 the problem [formula]has an infinite number of eigenfunctions on the level set ∫ΩM(|∇u|)=r, where M(t)=∫|t|0m(s)ds and g:R→R is odd satisfying some growth condition. Moreover, we show that the sequence of associated eigenvalues tends to infinity. We emphasize that no ▵2-condition is needed for M or for its conjugate, so the associated functionals are not continuously differentiable, in general