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Ground state solutions for a class of gauged Schrödinger equations with subcritical and critical exponential growth. (English) Zbl 1445.35142

Summary: We study a class of gauged nonlinear Schrödinger equations \[ \begin{cases} -\Delta u + \omega u + \lambda \left(\int_{|x|}^\infty \frac{h(s)}{s} u^2 (s) ds + \frac{h^2(|x|)}{|x|^2}\right) u = f(u) \text{ in } \mathbb{R}^2, \\ u \in H_r^1 (\mathbb{R}^2), \\ \end{cases} \] where \(\omega ,\lambda >0\) and \[ h(s) = \int_0^s \frac{r}{2} u^2 (r) dr. \] Under some suitable assumptions on \(f\) with critical exponential growth, we obtain a positive ground state solution by the non-Nehari manifold method. When \(f(u)\) has subcritical exponential growth, we prove the existence of a positive ground state solution by using a new approach. Our results generalize and improve the ones in [C. Ji and F. Fang, J. Math. Anal. Appl. 450, No. 1, 578–591 (2017; Zbl 1364.35296); J. Byeon et al., J. Funct. Anal. 263, No. 6, 1575–1608 (2012; Zbl 1248.35193)], and some other related literatures.

MSC:

35J20 Variational methods for second-order elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35B09 Positive solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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