Normalized Solutions of Nonautonomous Kirchhoff Equations: Sub- and Super-critical Cases

Abstract

AbstractIn this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2{\mathrm {d}}x\right) \Delta u-\lambda u=K(x)f(u), &{} x\in {\mathbb {R}}^3; \\ u\in H^1({\mathbb {R}}^3), \end{array} \right. \end{aligned}$$ - a + b ∫ R 3 | ∇ u | 2 d x Δ u - λ u = K ( x ) f ( u ) , x ∈ R 3 ; u ∈ H 1 ( R 3 ) , where $$a, b> 0$$ a , b > 0 , $$\lambda $$ λ is unknown and appears as a Lagrange multiplier, $$K\in {\mathcal {C}}({\mathbb {R}}^3, {\mathbb {R}}^+)$$ K ∈ C ( R 3 , R + ) with $$0<\lim _{|y|\rightarrow \infty }K(y)\le \inf _{{\mathbb {R}}^3} K$$ 0 < lim | y | → ∞ K ( y ) ≤ inf R 3 K , and $$f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})$$ f ∈ C ( R , R ) satisfies general $$L^2$$ L 2 -supercritical or $$L^2$$ L 2 -subcritical conditions. We introduce some new analytical techniques in order to exclude the vanishing and the dichotomy cases of minimizing sequences due to the presence of the potential K and the lack of the homogeneity of the nonlinearity f. This paper extends to the nonautonomous case previous results on prescribed $$L^2$$ L 2 -norm solutions of Kirchhoff problems.