## Two solutions for a planar equation with combined nonlinearities and critical growth

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## Abstract:

We prove the existence of two nonnegative nontrivial solutions for the equation \begin{equation*} -\Delta u -\frac {1}{2} (x\cdot \nabla u) = \lambda a(x)|u|^{q-2}u+f(u),\qquad x\in \mathbb {R}^2, \end{equation*} where $1<q<2$, $a$ is indefinite in sign, and the function $f(s)$ behaves like $e^{\alpha s^2}$ at infinity. The results hold for small values of the parameter $\lambda >0$.## References

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## Additional Information

**Marcelo F. Furtado**- Affiliation: Departamento de Matemática, Universidade de Brasília, 70910-900, Braília-Df, Brazil
- MR Author ID: 673056
- Email: mfurtado@unb.br
- Received by editor(s): January 23, 2019
- Published electronically: June 10, 2019
- Additional Notes: The author was partially supported by CNPq/Brazil and FAPDF/Brazil
- Communicated by: Joachim Krieger
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4397-4408 - MSC (2010): Primary 35J60; Secondary 35B33
- DOI: https://doi.org/10.1090/proc/14677
- MathSciNet review: 4002551