Existence Results and \(L^{\infty }\)-Bound of Solutions to Kirchhoff–Schrödinger–Hardy Type Equations Involving Double Phase Operators

This paper is concerned with several multiplicity results of nontrivial weak solutions to the Kirchhoff–Schrödinger type double phase problems with Hardy potentials. The main features of the paper are the presence of non-local Kirchhoff coefficients and the Hardy potential, the lack of compactness of the Euler–Lagrange functional, and the uniform boundedness for weak solutions. To establish such multiplicity results, we employ the fountain theorem and the dual fountain theorem as the primary tools, respectively. Finally, we demonstrate the \(L^{\infty }\) $L^{\infty }$ -bound for any possible weak solution by taking into consideration the De Giorgi iteration method and a truncated energy technique.