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=1NDj(aij(x,u)Diu)+12∑i,j=1NDtaij(x,u)DiuDju=f(x,u)+εg(x,u),x∈Ω,u∈H01(Ω), where Ω⊂RN(N≥3) is a bounded domain with smooth boundary and |ε| 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 εg(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 ε→0.