Critical Schrödinger–Poisson–Slater Equation With Lower Order Perturbation: Existence and Multiplicity of Solutions

ABSTRACT The Schrödinger‐Poisson‐Slater equation with zero mass is investigated in this paper. The nonlinearity has Sobolev critical growth with a lower order perturbation , and belongs to a suitable Lebesgue space. There are not so many works in the literature involving such problem, because that it has no linear term which is the so called zero mass case and the exponent belongs to the interval . Based on some key observations, we find a proper set where we shall find a nontrivial solution by constraining the corresponding energy functional on it. Under some technical conditions on the function , we further show that the solution obtained above is actually a ground state solution and any ground state solution of the above problem must be contained in such set and attain the infimum . By introducing a proper penalty function and employing some delicate analyses, we succeed in restoring the compactness for a modified functional under a precise threshold . Applying the critical point theorem established by Kajikiya [ J. Funct. Anal . 225: 352–370, 2005], we then obtain a sequence of solutions for above problem converging to zero. These results improve and complement some recent contributions in the literature.