Existence of multiple solutions of Schrödinger‐Kirchhoff‐type equations involving the p(.) ‐Laplacian in ℝN

Abstract

In this paper, we prove the existence of multiple solutions for the nonhomogeneous Schrödinger-Kirchhoff-type problem involving the p(.)-Laplacian { − (1 + b∫ R N 1 p(x) |∇u| p(x) dx) Δ p(x) u + V(x)|u| p(x)−2 u = (x, u) + g(x) in R N , u ∈ W 1,p(.) (R N) , where b ≥ 0 is a constant, N ≥ 2, Δ p(.) u ∶= div(|∇u| p(.)−2 ∇u) is the p(.)-Laplacian operator, p ∶ R N → R is Lipschitz continuous, V ∶ R N → R is a coercive type potential, ∶ R N × R → R and g ∶ R N → R functions verifying suitable conditions. We propose different assumptions on the nonlinear term ∶ R N × R → R to yield bounded Palais-Smale sequences and then prove that the special sequences we found converge to critical points, respectively. The solutions are obtained by the Mountain Pass Theorem, Ekeland variational principle, and Krasnoselskii genus theory.