Symmetric ground state solutions for the Choquard Logarithmic equation with exponential growth. (English) Zbl 1500.34013
Summary: We investigate the existence of ground state solutions for the fractional Choquard Logarithmic equation
\[
(- \Delta)^{1 / 2} u + V (x) u + (\ln | \cdot | \ast | u |^2) u = f (u), \quad x \in \mathbb{R},
\]
where \(V \in \mathcal{C} (\mathbb{R}, [ 0, \infty))\) and the \(f\) satisfies exponential critical growth. The present paper extends and complements the result of E. S. Böer and O. H. Miyagaki [“The Choquard logarithmic equation involving fractional Laplacian operator and a nonlinearity with exponential critical growth”, Preprint, arXiv:2011.12806]. In particular, our paper has two typical features. Firstly, using a weaker assumption on \(f\), we establish the energy inequality to exclude the vanishing case of the required Cerami sequence. Secondly, with the property of radial symmetry we shall use some new variational and analytic technique to establish our final result which is different to the arguments explored in [loc. cit.].
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Keywords:
Choquard logarithmic equation; critical exponential growth; Trudinger-Moser inequality; ground state solutionsReferences:
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