1 results
Soliton solutions for a class of Schrödinger equations with a positive quasilinear term and critical growth
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 18 February 2022, pp. 279-301
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- Article
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We consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics $-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$
in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms $h(s)$
with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$
when the parameter $\kappa >0$
is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.
