Multiple Solutions for a Critical Steklov Kirchhoff Equation

In the present work, we study some existing results related to a new class of Steklov p(x)-Kirchhoff problems with critical exponents. More precisely, we propose and prove some properties of the associated energy functional. In the first existence result, we use the mountain pass theorem to prove that the energy functional admits a critical point, which is a weak solution for such a problem. In the second main result, we use a symmetric version of the mountain pass theorem to prove that the investigated problem has an infinite number of solutions. Finally, in the third existence result, we use a critical point theorem proposed by Kajikiya to prove the existence of a sequence of solutions that tend to zero.