Existence results for a borderline case of a class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si29.svg" display="inline" id="d1e26"> <mml:mi>p</mml:mi> </mml:math> -Laplacian problems

The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically “linear” problem − div A 0 ( x ) + A ( x ) | u | p s | ∇ u | p − 2 ∇ u + s A ( x ) | u | p s − 2 u | ∇ u | p = μ | u | p ( s + 1 ) − 2 u + g ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N , N ≥ 2 , 1 < p < N , s > 1 / p , both the coefficients A 0 ( x ) and A ( x ) are in L ∞ ( Ω ) and far away from 0, μ ∈ R , and the “perturbation” term g ( x , t ) is a Carathéodory function on Ω × R which grows as | t | r − 1 with 1 ≤ r < p ( s + 1 ) and is such that g ( x , t ) ≈ ν | t | p − 2 t as t → 0 . By introducing suitable thresholds for the parameters ν and μ , which are related to the coefficients A 0 ( x ) , respectively A ( x ) , under suitable hypotheses on g ( x , t ) , the existence of a nontrivial weak solution is proved if either ν is large enough with μ small enough or ν is small enough with μ large enough. Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used.