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In this paper we consider the $\varphi$-Laplacian problem with Dirichlet boundary condition, $$-\mathrm{div}\left(\varphi \left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right){=\lambda g(·)\varphi \left(u\right)\text{in}\Omega ,\lambda \in ℝ\text{and}u|}_{\partial \Omega }=0.$$ The term $\varphi$ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty }\left(\Omega \right)$ and $\Omega \subseteq {ℝ}^N$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer...

In this paper we study equivalence classes of generic $1$-parameter germs of real analytic families ${𝒬}_{\epsilon }$ unfolding codimension $1$ germs of diffeomorphisms ${𝒬}_0:(ℝ,0)\to (ℝ,0)$ with a fixed point at the origin and multiplier $-1,$ under (weak) analytic conjugacy. These germs are generic unfoldings of the flip bifurcation. Two such germs are analytically conjugate if and only if their second iterates, ${𝒫}_{\epsilon }={𝒬}_{\epsilon }^{\circ 2},$ are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding of the Ecalle...