TY  - JOUR
ID  - Zhang2026
LB  - Zhang2026
AU  - Zhang, Limin
AU  - Yuan, Shuai
TI  - Ground states for the planar indefinite Choquard equation with supercritical exponential growth
ST  - Ground states for the planar indefinite Choquard equation with supercritical exponential growth
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2026
VL  - 46
IS  - 4
SP  - 563
EP  - 592
DO  - https://doi.org/10.7494/OpMath.202607081
UR  - https://doi.org/10.7494/OpMath.202607081
AB  - This paper investigates a class of strongly indefinite Choquard equations formulated on the two-dimensional unit ball \(B\). Specifically, consider the following equation: \[\begin{cases}-\Delta u+V(x)u=(I_{\mu}\ast F(u))f(u), &x\in B, \\ u\in H^1_{0,\mathrm{rad}}(B), & \end{cases}\] where \(\mu\in(0,2)\) and \(I_{\mu}\) represents the classical Riesz potential. A fundamental hypothesis is that zero lies within a spectral gap of the Schrödinger operator \(-\Delta+V\). Furthermore, the continuous nonlinearity \(f(t)\) is characterized by a supercritical exponential growth governed by \(\exp[(\beta+|x|^{\alpha})t^{2}]\) with \(\beta, \alpha\gt 0\). Most existing works on this problem are limited to the subcritical or critical exponential growth cases. However, dealing with supercritical growth is much more difficult, especially when estimating the exact upper bound of the minimax levels. To overcome this, we use an approximation method and fine estimates to prove that this indefinite problem has a positive ground state solution. Our approach can also be applied to other elliptic problems with supercritical growth.
KW  - strongly indefinite
KW  - Trudinger-Moser inequality
KW  - supercritical exponential growth
SN  - 1232-9274
ER  -