(p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group

Abstract

The paper deals with the existence of solutions for (p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in {{\mathbb{H}}}^{n} .

On the noncooperative Schrödinger–Kirchhoff system involving the critical nonlinearities on the Heisenberg group

“…After that, using a minimization argument and the Ekeland variational principle, Lam and Lu [13] considered that in the absence of perturbation (ε = 0), and they obtained the existence and multiplicity of solutions of problem (1.3). Subsequently, by variational methods, Pucci et al [25,26] dealt with the existence of nontrivial solutions for (p, Q) equations and (p, Q) system in the Heisenberg group H n by considering the case without the potential function.…”

Existence and multiplicity of solutions for critical Kirchhoff-Choquard equations involving the fractional p-Laplacian on the Heisenberg group

“…Moreover, the existence of nontrivial solutions were obtained under non-degenerate and degenerate conditions. For more fascinating results, we refer to An and Liu [1], Bordoni and Pucci [3], Liu et al [13], Liu and Zhang [14], Pucci [16,17], and Pucci and Temperini [18,19].…”

Ground state solutions for planar coupled system involving nonlinear Schrödinger equations with critical exponential growth

“…In particular, it plays an important role in studying the existence of nontrivial solutions of many Laplace‐type nonlinear elliptic boundary value problems. To the best of our knowledge, for many Laplace‐type nonlinear elliptic boundary value problems, it seems necessary to use (WN)‐condition to obtain the Nehari‐type ground state solutions; see, for example, Alves, Cassani, Tarsi and Yang, 18 Alves and Germano, 19 Alves and Souto, 20 Alves, Souto and Montenegro, 21 Chen, Fiscella, Pucci and Tang, 22 Chen and Tang, 23,24 de Figueiredo, Miyagaki and Ruf, 25 Mingqi, Rădulescu and Zhang, 26‐28 Tang and Cheng, 29 Tang and Chen, 30 Tang, Chen, Lin and Yu, 31 Tang and Lin, 32 Xiang, Zhang and Rădulescu, 33 Pucci and Temperini, 34,35 and Pucci and Vitillaro 36 …”