DOI: 10.1515/gmj-2026-3053 ISSN: 1072-947X

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

Ouarda Saifia, Samah Horrigue
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Abstract

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 ) {\gamma\in(-1,1)}
, whose sign induces two fundamentally different variational structures. For
γ > 0 {\gamma>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 {\gamma<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.

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