Existence and multiplicity of solutions for Kirchhoff–Schrödinger type equations involving p(x)-Laplacian on the entire space R^N

This study is concerned with the following elliptic equation: −M(∫_R^N [Formula presented]|∇u|^p(x)dx)div(|∇u|^p(x)−2∇u)+V(x)|u|^p(x)−2u=λf(x,u)inR^N,where M∈C(R⁺) is a Kirchhoff-type function, the potential function V:R^N→(0,∞) is continuous, and f:R^N×R→R satisfies a Carathéodory condition. The aim is to determine the precise positive interval of λ for which the problem admits at least two nontrivial solutions by using abstract critical point results for an energy functional satisfying the Cerami condition. It should be noted that the existence of at least one nontrivial weak solution is established by employing the mountain pass theorem. Moreover, the existence of an unbounded sequence of nontrivial weak solutions follows from the fountain theorem owing to the variational nature of the problem.