Necessary and sufficient conditions for ground state solutions to planar Kirchhoff-type equations

In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem: \[ -(1+b∫R^2|∇ u|^2\,{ d}x)Δ u+ω u=|u|ᵖ⁻²u, x^2. \] where b,\, ω >0 are constants, $p>2$ . Based on variational methods, regularity theory and Schwarz symmetrization, the equivalence of ground state solutions for the above problem with the minimizers for some minimization problems is obtained. In particular, a new scale technique, together with Lagrange multipliers, is delicately employed to overcome some intrinsic difficulties.