Limit Property of an L2-Normalized Solution for an L2-Subcritical Kirchhoff-Type Equation with a Variable Exponent - Xincai Zhu; Hanxiao Wu (Axioms)
This paper is concerned with the following <semantics>L2</semantics>-subcritical Kirchhoff-type equation <semantics>−<mfenced separators="" open="(" close=")">a+b<mfenced separators="" open="(" close=")">∫R2|∇u|2dx</mfenced>s</mfenced>Δu+V(x)u=μu+β|u|2u,<mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace>x∈R2</semantics>, with <semantics>∫R2|u|2dx=1</semantics>. We give a detailed analysis of the limit property of the <semantics>L2</semantics>-normalized solution when exponent s tends toward 0 from the right (i.e., <semantics>s↘0</semantics>). Our research extends previous works, in which the authors have displayed the limit behavior of <semantics>L2</semantics>-normalized solutions when <semantics>s=1</semantics> as <semantics>a↘0</semantics> or <semantics>b↘0</semantics>.