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

Furtado, Marcelo

doi:10.1090/proc/14677

Two solutions for a planar equation with combined nonlinearities and critical growth
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by Marcelo F. Furtado

Proc. Amer. Math. Soc. 147 (2019), 4397-4408

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 \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 $\lambda >0$.

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