Uniqueness results for mixed local and nonlocal equations with singular nonlinearities and source terms

Abstract This paper considers a local and nonlocal problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: where is an open bounded domain with a boundary and We assume that and , with the conditions or , corresponding to the homogeneous and nonhomogeneous cases, respectively. The parameters satisfy and . The function is nonzero and belongs to a suitable Lebesgue space for some , or satisfies a growth condition involving negative powers of the distance function near the boundary . Additionally, is a positive function defined within appropriate Lebesgue spaces. The primary objectives of this paper are twofold. First, we establish the uniqueness of infinite energy solutions to problem () by introducing a novel comparison principle under certain conditions. Second, we derive several existence results for weak solutions in various senses, accompanied by regularity results for problem (). Furthermore, we present a nonexistence result when the function and is near the boundary, under the condition . Our approach leverages the Picone identities on one hand and the interaction between the local and nonlocal terms on the other hand.