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Planar Kirchhoff equations with critical exponential growth and trapping potential


We are concerned with the following Kirchhoff equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( a+ b\int _{\mathbb {R}^2}|\nabla u|^2\mathrm {d}x\right) \Delta u+V(x)u=f(u), \;\;&{} \text{ in } \ \ \mathbb {R}^2,\\ u\in H^1(\mathbb {R}^2), \end{array}\right. } \end{aligned}$$

where \(a,\, b\) are positive constants, \(V\in \mathcal {C}(\mathbb {R}^2, (0,\infty ))\) is a trapping potential, and f has critical exponential growth of Trudinger–Moser type. By developing some new analytical approaches and techniques, we prove the existence of nontrivial solutions and least energy solutions. Without any monotonicity conditions on f, we also give the mountain pass characterization of the least energy solution by constructing a fine path. In particular, we remove the common restriction on \(\liminf _{t\rightarrow +\infty }\frac{tf(t)}{e^{\alpha _0 t^2}}\), which is crucial in the literature to overcome the loss of the compactness caused by the critical exponential nonlinearity. Our approach could be extended to other classes of critical exponential growth problems with trapping potentials.

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This work is partially supported by the National Natural Science Foundation of China (No. 11971485, No. 12001542), Hunan Provincial Natural Science Foundation (No. 2021JJ40703) and Innovation Project of Graduate Students of Central South University (No.1053320213251). The author Lixi Wen acknowledges the financial support of the China Scholarship Council (No. 202006370225) and would like to thank also the Embassy of the People’s Republic of China in Romania. The research of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, project number PCE 137/2021, within PNCDI III.

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Chen, S., Rădulescu, V.D., Tang, X. et al. Planar Kirchhoff equations with critical exponential growth and trapping potential. Math. Z. 302, 1061–1089 (2022).

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  • Kirchhoff equation
  • Critical exponential growth
  • Trudinger–Moser inequality
  • Trapping potential

Mathematics Subject Classification

  • 35J20
  • 35J62
  • 35Q55