In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈ℝ×H^(1)(ℝ^(N))to the general Kirchhoff problem-M\left(\int_{\mathbb{R}^N}\vert\nabla u\ve...In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈ℝ×H^(1)(ℝ^(N))to the general Kirchhoff problem-M\left(\int_{\mathbb{R}^N}\vert\nabla u\vert^2{\rm d}x\right)\Delta u+\lambda u=g(u)~\hbox{in}~\mathbb{R}^N,u\in H^1(\mathbb{R}^N),N\geq 1,satisfying the normalization constraint\int_{\mathbb{R}^N}u^2{\rm d}x=c,where M∈C([0,∞))is a given function satisfying some suitable assumptions.Our argument is not by the classical variational method,but by a global branch approach developed by Jeanjean et al.[J Math Pures Appl,2024,183:44–75]and a direct correspondence,so we can handle in a unified way the nonlinearities g(s),which are either mass subcritical,mass critical or mass supercritical.展开更多
In this paper,based on some prior estimates,we show that the essential spectrum λ=0 is a bifurcation point for a superlinear elliptic equation with only local conditions,which generalizes a series of earlier results ...In this paper,based on some prior estimates,we show that the essential spectrum λ=0 is a bifurcation point for a superlinear elliptic equation with only local conditions,which generalizes a series of earlier results on an open problem proposed by Stuart(1983).展开更多
基金supported by the NSFC(12271184)the Guangzhou Basic and Applied Basic Research Foundation(2024A04J10001).
文摘In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈ℝ×H^(1)(ℝ^(N))to the general Kirchhoff problem-M\left(\int_{\mathbb{R}^N}\vert\nabla u\vert^2{\rm d}x\right)\Delta u+\lambda u=g(u)~\hbox{in}~\mathbb{R}^N,u\in H^1(\mathbb{R}^N),N\geq 1,satisfying the normalization constraint\int_{\mathbb{R}^N}u^2{\rm d}x=c,where M∈C([0,∞))is a given function satisfying some suitable assumptions.Our argument is not by the classical variational method,but by a global branch approach developed by Jeanjean et al.[J Math Pures Appl,2024,183:44–75]and a direct correspondence,so we can handle in a unified way the nonlinearities g(s),which are either mass subcritical,mass critical or mass supercritical.
基金supported by National Natural Science Foundation of China(Grant Nos.11801581,11871123,11931012 and 12271184)Guangdong Basic and Applied Basic Research Foundation(Grant Nos.2021A1515010034 and 2018A030310082)+2 种基金Guangzhou Association for Science and Technology(Grant No.202102020225)Chongqing Science and Technology Bureau(Grant No.JDDSTD201802)Chongqing University Science Foundation(Grant No.CXQT21021).
文摘In this paper,based on some prior estimates,we show that the essential spectrum λ=0 is a bifurcation point for a superlinear elliptic equation with only local conditions,which generalizes a series of earlier results on an open problem proposed by Stuart(1983).
