Elliptic equations in RN involving Sobolev supercritical and upper Hardy-Littlewood-Sobolev critical exponents

This paper deals with the existence of nontrivial nonnegative solutions of model parametric elliptic equations in RN involving a possibly supercritical term in Sobolev sense, and a nonlocal term with the upper Hardy-Littlewood-Sobolev critical exponent. Under some conditions, we describe the precise parametric range of existence and nonexistence of a nonnegative solution. Furthermore, in a slightly smaller range, a second nontrivial nonnegative solution is constructed. Additionally, infinitely many solutions, with energy asymptotic behavior, are also obtained if the growth near the origin is concave. These results, which are inspired by the pioneering work of Alama and Tarantello (1996) [2] in the local case of Dirichlet problems in bounded domains, are obtained by combining variational methods, Leray-Schauder degree theory, and the Krasnoselskii genus via biorthogonal functionals in separable and reflexive Banach spaces.<br /> (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.