Necessary and sufficient conditions for groundstate 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 integral(R2)|del u|(2)dx)Delta u+omega u=|u|(p-2)u, x is an element of 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.