Study of a ϕ -Hilfer fractional Schrödinger–Choquard problem via variational methods

We study a nonlinear boundary value problem involving ϕ-Hilfer fractional derivatives and a nonlocal Riesz-type integral term on a smooth bounded domain. The equation contains a power-type nonlinearity with exponent γ ∈ ( - 1 , 1 ) {γ∈(-1,1)} , whose sign induces two fundamentally different variational structures. For γ > 0 {γ>0} , the associated energy functional exhibits a mountain-pass geometry, and we establish the existence of nontrivial weak solutions via the Mountain Pass Theorem. In contrast, when γ < 0 {γ<0} , the singular behavior of the nonlinearity leads to a coercive but non-regular functional, for which we construct an appropriate min–max framework to obtain solutions. The analysis provides a unified treatment of regular and singular nonlinearities in fractional problems driven by ϕ-Hilfer operators and extends existing theories for nonlocal fractional equations with integral interactions.