Singular Double Phase Kirchhoff Type Problem with a General Nonlocal Integrodifferential Operator

In this study, we study a singular double-phase Kirchhoff problem involving a fractional nonlocal integrodifferential operator. More precisely, we reformulate the studied problem into an equivalent integral equation and derive the corresponding energy functional. By combining a variational method with monotonicity techniques, we establish that this functional admits a minimum point in an appropriate Sobolev space. However, due to the presence of the singular term, this minimum does not necessarily correspond to a critical point of the function. For this, we use the implicit function theorem to prove that this minimum corresponds to a weak solution for such a problem. To validate our main results, an example is presented.