A Critical Concave–Convex Kirchhoff-Type Equation in r4 Involving Potentials Which May Vanish at Infinity

We establish the existence and multiplicity of solutions for a Kirchhoff-type problem in R4 involving a critical and concave–convex nonlinearity. Since in dimension four, the Sobolev critical exponent is 2 ∗= 4 , there is a tie between the growth of the nonlocal term and the critical nonlinearity. This turns out to be a challenge to study our problem from the variational point of view. Some of the main tools used in this paper are the mountain-pass and Ekeland’s theorems, Lions’ Concentration Compactness Principle and an extension to RN of the Struwe’s global compactness theorem. © 2021, Springer Nature Switzerland AG.