Infinitely many solutions for p ‐Kirchhoff equation with concave–convex nonlinearities in R N

In this paper, we study the existence of infinitely many solutions to p ‐Kirchhoff‐type equation 0.1a + μ∫R N( | ∇ u | p + V ( x ) | u | p ) dxτ( − Δ p u + V ( x ) | u | p − 2 u ) = f ( x , u ) ,x ∈ R N ,where f ( x , u ) = λ h 1 ( x )| u | m − 2 u + h 2 ( x )| u | q − 2 u , a ≥0, μ > 0, τ > 0, λ ≥0 and 1 < m < p < N , p < q < p * = pN N − p . The potential function V ( x ) ∈ C ( R N ) verifies 0 < inf x ∈ R NV ( x ) ≤ sup x ∈ R NV ( x ) < ∞ , and h 1 ( x ), h 2 ( x ) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ 0 >0 such that problem admits infinitely many nonnegative high‐energy solutions provided that λ ∈[0, λ 0 ) and p ( τ + 1 ) < q < p * = pN N − p . Also, we prove that problem has at least a nontrivial solution under the assumption f ( x , u ) = h 2 | u | q − 2 u , p < q < min{ p *, p ( τ + 1)} and has infinitely many nonnegative solutions for f ( x , u ) = h 1 | u | m − 2 u ,1 < m < p . Copyright © 2015 John Wiley & Sons, Ltd.