Multiple Solutions to p ‐Laplacian Problems with Asymptotic Nonlinearity as u p −1 at Infinity

The paper studies the existence of multiple solutions to the following p ‐Laplacian type elliptic problem ( p > 1):{− Δ p u ( x ) ≡ :  div (| ∇ u |p − 2 ∇ u ) = f ( x , u )      x ∈ Ωu ∈   W 0 1 , p( Ω ) ,where Ω is a bounded domain in R N ( N ⩾ 1) with smooth boundary ∂Ω, and f ( x , u ) goes asymptotically in u to ∣ u ∣ p −2 u at infinity. It is well known that this kind of nonlinear term creates some difficulties in the application of the mountain pass theorem because of the lack of an Ambrosetti–Rabinowitz type superlinear condition on f ( x , u ). An improved mountain pass theorem is used to prove that the above problem possesses multiple solutions under some natural conditions on f ( x , u ), and some known results are generalized.