Convergence properties of a $\varphi$-Laplacian under natural constraints

Abstract

In this paper, we consider the $\varphi$-Laplacian problem with Dirichlet boundary condition,

The term $\varphi$ is a real odd and increasing homeomorphism, $g$ is a nontrivial, nonnegative function in $L^{\infty }(\Omega )$, and $\Omega \subset {\mathbb{R}}^N$ is a bounded connected domain. We examine the asymptotic behavior of sequences of eigenvalues of the differential equation. The treatment is based solely on the asymptotic homogeneity of $\varphi$ plus an additional condition on the domain (the segment property). We choose a sequence of eigenfunctions tending either to zero or infinity (in the sense of the norm). The core result of these notes shows that the liminf of the associated sequence of eigenvalues coincides with the first eigenvalue of the usual $p$-Laplace operator, and that the weak-$\star$ limit of the corresponding eigenfunctions is an associated ground state.