An elliptic problem involving critical Choquard and singular discontinuous nonlinearity

Abstract The present article investigates the existence, multiplicity and regularity of weak solutions of problem involving a combination of critical Hartree-type nonlinearity along with singular and discontinuous nonlinearities (see <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="script">P</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $\left({\mathcal{P}}_{\lambda }\right)$ below). By applying variational methods and using the notion of generalized gradients for Lipschitz continuous functional, we obtain the existence and the multiplicity of weak solutions for some suitable range of λ and γ . Finally by studying the L ∞ -estimates and boundary behaviour of weak solutions, we prove their Hölder and Sobolev regularity.