English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, we consider the existence and asymptotic properties ofsolutions to the following Kirchhoff equation\begin{equation}\label{1}\nonumber - \Bigl(a+b\int_{{\R^3}} {{{\left| {\nablau} \right|}^2}}\Bigl) \Delta u  =\lambda u+ {| u |^{p - 2}}u+\mu {| u |^{q - 2}}u \text { in } \mathbb{R}^{3}\end{equation} under the normalized constraint $\int_{{\mathbb{R}^3}}{{u}^2}=c^2$, where $a\!>\!0$, $b\!>\!0$, $c\!>\!0$,$2\!<\!q\!<\!\frac{14}{3}\!<\! p\!\leq\!6$ or $\frac{14}{3}\!<\!q\!<\!p\!\leq\! 6$, $\mu\!>\!0$ and $\lambda\!\in\!\R$ appears as a Lagrangemultiplier. In both cases for the range of $p$ and $q$, the Sobolev criticalexponent $p\!=\!6$ is involved and the corresponding energy functional isunbounded from below on $S_c=\Big\{ u \in H^{1}({\mathbb{R}^3}):\int_{{\mathbb{R}^3}} {{u}^2}=c^2 \Big\}$. If $2\!<\!q\!<\!\frac{10}{3}$ and$\frac{14}{3}\!<\! p\!<\!6$, we obtain a multiplicity result to the equation.If $2\!<\!q\!<\!\frac{10}{3}\!<\! p\!=\!6$ or $\frac{14}{3}\!<\!q\!<\!p\!\leq\! 6$, we get a ground state solution to the equation. Furthermore, wederive several asymptotic results on the obtained normalized solutions.  Our results extend the results of N. Soave (J. Differential Equations 2020$\&$ J. Funct. Anal. 2020), which studied the nonlinear Schr\"{o}dingerequations with combined nonlinearities, to the Kirchhoff equations. To dealwith the special difficulties created by the nonlocal term $({\int_{{\R^3}}{\left| {\nabla u} \right|} ^2}) \Delta u$ appearing in Kirchhoff typeequations, we develop a perturbed Pohozaev constraint approach and we find away to get a clear picture of the profile of the fiber map via carefulanalysis. In the meantime, we need some subtle energy estimates under the$L^2$-constraint to recover compactness in the Sobolev critical case.

Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent