Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case

Abstract

Abstract We study the following nonlinear mass supercritical Kirchhoff equation: − ( a + b ∫ R N | ∇ u | 2 ) △ u + μ u = f ( u ) in R N , $$ - \biggl(a+b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2} \biggr) \triangle u+ \mu u=f(u) \quad \text{in } {\mathbb{R}^{N}}, $$ where a , b , m > 0 $a ,b,m>0$ are prescribed, and the normalized constrain ∫ R N | u | 2 d x = m $\int _{\mathbb{R}^{N}}|u|^{2}\,dx =m$ is satisfied in the case 1 ≤ N ≤ 3 $1\leq N\leq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when 1 ≤ N ≤ 3 $1\leq N\leq 3$ and obtain infinitely many radial solutions when 2 ≤ N ≤ 3 $2\leq N\leq 3$ by constructing a particular bounded Palais–Smale sequence.

Keywords

• Normalized ground states
• Kirchhof-type problems
• Multiplicity results
• Global minimizers