The Existence And Asymptotic Properties Of Nontrivial Solutions Of Nonlinear 2-q-Laplacian Type Problems With Linking Geometry

In this paper, we study the existence of nontrivial solutions and the asymptotic properties of solutions to the following nonlinear elliptic problems with singular perturbation in a bounded domainΩ(?) R^N:whereε≥0,â–³_qu = div(?), 1 < q < 2 < N, and thatα∈(?), f∈C⁰((?)×R¹, R¹), f(x,t) is super-linear near t = 0 and subcritical near t =∞, together with the Ambrosetti - Rabinowitz conditions and some technical conditions, so that (1.1)_εpossesses the "Linking Geometric Structure" . We prove that there exists anε₀ > 0 such that for any 0≤ε≤ε₀, (1.1)_εpossesses at least a nontrivial solution u_εand for any sequenceε_n withε_n→0⁺, there is a subsequence (?) such that (?) in H₀¹(Ω) for some nontrivial solution (?) of (1.1)₀. Our result generalizes the classical result in [10] about the existence of nontrivial solutions of (1.1)₀ to the 2 - q - Laplacian type problem (1.1)_ε.