A study of extremal parameter for fractional singular Choquard problem

In this work, we study the singular problem involving fractional Laplacian operator perturbed with a Choquard nonlinearity using the idea of constrained minimization based on Nehari manifold. Precisely, for some ε>0$\epsilon >0$, we have proved the existence of two solutions when the parameter λ∈(0,λ∗+ε)$\lambda \in (0, \lambda ^*+\epsilon )$, adding to the existing works dealing with multiplicity of solutions when the parameter λ strictly lies below λ∗$\lambda ^*$. We have given a variational characterization of the parametric value λ∗$\lambda ^*$, which is an extremal value of the parameter λ involved in the problem up to which the Nehari manifold method can be applied successfully. The paper highlights a fine analysis via fibering maps even for λ≥λ∗$\lambda \ge \lambda ^*$ to establish an existence of two different positive solutions for the underlying problem.