Existence of nontrivial solutions for quasilinear elliptic equations at critical growth

Abstract

Abstract In this paper, we study the existence of nontrivial weak solutions for the following quasilinear elliptic equation with natural growth - D j ( a ij ( x , u ) D i u ) + 1 2 ∂ s a ij ( x , u ) D i uD j u = f ( x , u ) , in a bounded smooth domain Ω ⊂ R N , where aij(x, u) are functions possibly unbounded with respect to u.