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


Author: Marcelo F. Furtado
Journal: Proc. Amer. Math. Soc. 147 (2019), 4397-4408
MSC (2010): Primary 35J60; Secondary 35B33
DOI: https://doi.org/10.1090/proc/14677
Published electronically: June 10, 2019
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Abstract: We prove the existence of two nonnegative nontrivial solutions for the equation

$\displaystyle -\Delta u -\frac {1}{2} (x\cdot \nabla u) = \lambda a(x)\vert u\vert^{q-2}u+f(u),\qquad x\in \mathbb{R}^2,$    

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$.

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Additional Information

Marcelo F. Furtado
Affiliation: Departamento de Matemática, Universidade de Brasília, 70910-900, Braília-Df, Brazil
Email: mfurtado@unb.br

DOI: https://doi.org/10.1090/proc/14677
Keywords: Concave-convex problems, critical exponential growth, Trudinger-Moser inequality
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
Article copyright: © Copyright 2019 American Mathematical Society