Multiplicity and asymptotic behavior of solutions for quasilinear elliptic equations with small perturbations

Abstract This paper considers the following general form of quasilinear elliptic equation with a small perturbation: { − ∑ i , j = 1 N D j ( a i j ( x , u ) D i u ) + 1 2 ∑ i , j = 1 N D t a i j ( x , u ) D i u D j u = f ( x , u ) + e g ( x , u ) , x ∈ Ω , u ∈ H 0 1 ( Ω ) , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary and | e | small enough. We assume the main term in the equation to have a mountain pass structure but do not suppose any conditions for the perturbation term e g ( x , u ) . Then we prove the equation possesses a positive solution, a negative solution and a sign-changing solution. Moreover, we are able to obtain the asymptotic behavior of these solutions as e → 0 .