Extension of a result by Lindquist to Lebesgue spaces with variable exponents

We investigate the family of variable-exponent type eigenvalue problems −Δpj(⋅)u=λj|u|p(⋅)−2u−Δpj(⋅)u=λj|u|p(⋅)−2u and prove the convergence of the eigenfunctions under suitable conditions on the sequence (pj)(pj).