On the planar supercritical Kirchhoff problem

In this paper, we investigate the planar Kirchhoff problem with supercritical nonlinearity [Formula: see text], which behaviors as [Formula: see text] and [Formula: see text] with [Formula: see text], [Formula: see text]. The supercritical nature of the nonlinear term leads to excessive energy of the solution or makes it difficult to estimate effectively, thereby rendering the standard energy estimation methods applicable in subcritical and critical cases ineffective. We develop some new approaches to estimate precisely the minimax level of the energy functional and to prove the mentioned problem has a Nehari-type ground state solution. In particular, if the weighted term [Formula: see text] degenerates to a nonzero constant, we can obtain the constant exponent supercritical the ones in [C. O. Alves and L. J. Shen, On existence of solutions for some classes of elliptic problems with supercritical exponential growth, Math. Z. 306(2) (2024) 29], rather than variable exponent, which are more complex because they usually do not inherit the excellent properties of the standard constant exponential function space.