Properties of minimizers for L2-subcritical Kirchhoff energy functionals

Abstract

We consider the properties of minimizers for the following constraint minimization problem: i(c)≔infu∈S1Ic(u),i\left(c):= \mathop{\inf }\limits_{u\in {S}_{1}}{I}_{c}\left(u), where the L2{L}^{2}-unite sphere S1={u∈H1(RN)∣∫RNV(x)u2dxc˜pp∈0,4Nc\gt {\widetilde{c}}_{p}\hspace{0.33em}\left(\phantom{\rule[-0.75em]{}{0ex}},p\in \left(\phantom{\rule[-0.75em]{}{0ex}},0,\frac{4}{N}\right]\right), or c≥c˜pp∈4N,8Nc\ge {\widetilde{c}}_{p}\hspace{0.33em}\left(\phantom{\rule[-0.75em]{}{0ex}},p\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{4}{N},\frac{8}{N}\right)\right); if V(x)≢0V\left(x)\not\equiv 0, the concentration behavior of minimizers for i(c)i\left(c) is discussed as c→+∞.c\to +\infty . Moreover, the uniqueness of minimizers for i(c)i\left(c) is also proved as cc large enough, which extends the related results.

Keywords

• subcritical
• kirchhoff energy functional
• constrained variational method
• energy estimates
• local uniqueness
• 35j20
• 35j60
• 47j30