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Normalized solutions for some quasilinear elliptic equation with critical Sobolev exponent (2306.10207v1)

Published 16 Jun 2023 in math.AP and math.FA

Abstract: Consider the equation \begin{equation*} -\Delta_p u =\lambda |u|{p-2}u+\mu|u|{q-2}u+|u|{p\ast-2}u\ \ {\rm in}\ \RN \end{equation*} under the normalized constraint $$\int_{ \RN}|u|p=cp,$$ where $-\Delta_pu={\rm div} (|\nabla u|{p-2}\nabla u)$, $1<p<N$, $p<q<p^\ast=\frac{Np}{N-p}$, $c,\mu\>0$ and $\lambda\in\R$. In the purely $Lp$-subcritical case, we obtain the existence of ground state solution by virtue of truncation technique, and obtain multiplicity of normalized solutions. In the purely $Lp$-critical and supercritical case, we drive the existence of positive ground state solution, respectively. Finally, we investigate the asymptotic behavior of ground state solutions obtained above as $\mu\to0+$.

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