Abstract. In this paper, we deal with the following elliptic type problem ⎩⎨⎧(−Δ)q(.)s(.)u+λVu=α∣u∣p(.)−2u+β∣u∣k(.)−2uu=0 in Ω, in Rn\Ω,
where q(.):Ω×Ω→R is a measurable function and s(.):Rn×Rn→(0,1) is a continuous function, n>q(x,y)s(x,y) for all (x,y)∈Ω×Ω,(−Δ)q(.)s(.) is the variable-order fractional Laplace operator, and V is a positive continuous potential. Using the mountain pass category theorem and Ekeland’s variational principle, we obtain the existence of a least two different solutions for all λ>0. Besides, we prove that these solutions converge to two of the infinitely many solutions of a limit problem as λ→+∞.
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