A double-phase eigenvalue problem with large exponents

Abstract

In the present article, we consider a double-phase eigenvalue problem with large exponents. Let λ(pn,qn)1{\lambda }_{\left({p}_{n},{q}_{n})}^{1} be the first eigenvalues and un{u}_{n} be the first eigenfunctions, normalized by ‖un‖ℋn=1\Vert {u}_{n}{\Vert }_{{{\mathcal{ {\mathcal H} }}}_{n}}=1. Under some assumptions on the exponents pn{p}_{n} and qn{q}_{n}, we show that λ(pn,qn)1{\lambda }_{\left({p}_{n},{q}_{n})}^{1} converges to Λ∞{\Lambda }_{\infty } and un{u}_{n} converges to u∞{u}_{\infty } uniformly in the space Cα(Ω){C}^{\alpha }\left(\Omega ), and u∞{u}_{\infty } is a nontrivial viscosity solution to a Dirichlet ∞\infty -Laplacian problem.

Keywords

• musielak-orlicz spaces
• viscosity solution
• double-phase problem
• ∞-laplacian
• eigenvalue
• 46e30
• 35d40
• 35p30