The asymptotic behavior of solutions for a perturbed system of eigenvalue problems

Abstract In this article we study the existence of solutions for a family of perturbed eigenvalue problems described by a system consisting of two partial differential equations in which the differential operators involved are the sum between a p -Laplacian and a q -Laplacian with $$q\ne p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> but depending on p . Next, we study the asymptotic behavior, as $$p\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , of the sequence of solutions and we show that, passing eventually to a subsequence, it converges uniformly to a certain limit given by a pair of continuous functions. Moreover, we identify the limiting equations which have as solutions the limiting functions.