Nodal solutions to ( p , q )-Laplacian equations with critical growth

Abstract

In this paper, a class of ( p , q )-Laplacian equations with critical growth is taken into consideration: − Δ p u − Δ q u + ( | u | p − 2 + | u | q − 2 ) u + λ ϕ | u | q − 2 u = μ g ( u ) + | u | q ∗ − 2 u , x ∈ R 3 , − Δ ϕ = | u | q , x ∈ R 3 , where Δ ξ u = div ( | ∇ u | ξ − 2 ∇ u ) is the ξ-Laplacian operator ( ξ = p , q ), 3 2 < p < q < 3, λ and μ are positive parameters, q ∗ = 3 q / ( 3 − q ) is the Sobolev critical exponent. We use a primary technique of constrained minimization to determine the existence, energy estimate and convergence property of nodal (that is, sign-changing) solutions under appropriate conditions on g, and thus generalize the existing results.