A class of -Laplacian type equation with concave–convex nonlinearities in bounded domain

In this paper, our main purpose is to establish the existence of multiple solutions of a class of p – q -Laplacian equation involving concave–convex nonlinearities: { − Δ p u − Δ q u = θ V ( x ) | u | r − 2 u + | u | p ⁎ − 2 u + λ f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω where Ω is a bounded domain in R N , λ , θ > 0 , 1 < r < q < p < N and p ⁎ = N p N − p is the critical Sobolev exponent, Δ s u = div ( | ∇ u | s − 2 ∇ u ) is the s-Laplacian of u. We prove that for any λ ∈ ( 0 , λ ⁎ ) , λ ⁎ > 0 is a constant, there is a θ ⁎ > 0 , such that for every θ ∈ ( 0 , θ ⁎ ) , the above problem possesses infinitely many weak solutions. We also obtain some results for the case 1 < q < p < r < p ⁎ . The existence results of solutions are obtained by variational methods.