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Normalized solutions to the quasilinear Schrödinger equations with combined nonlinearities
Part of:
Elliptic equations and systems
Published online by Cambridge University Press: 12 April 2024
Abstract
We consider the radially symmetric positive solutions to quasilinear problem
\begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*}
having prescribed mass 
$\int_{\mathbb{R}^{N}}|u|^2 =a^2,$ where a > 0 is a constant, λ appears as a Lagrange multiplier. We focus on the pure L2-supercritical case and combination case of L2-subcritical and L2-supercritical nonlinearities
\begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*}
where 
$\bar{p}:=4+\frac{4}{N}$ is the L2-critical exponent. Our work extends and develops some recent results in the literature.
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- © The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.
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