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:\[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \]where $b,\, \omega >0$ 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.