Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4

Abstract

In this article, we study the following Kirchhoff equation with combined nonlinearities: −a+b∫R4∣∇u∣2dxΔu+λu=μ∣u∣q−2u+∣u∣2u,inR4,∫R4∣u∣2dx=c2,\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu {| u| }^{q-2}u+{| u| }^{2}u,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{4},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| u| }^{2}{\rm{d}}x={c}^{2},\end{array}\right. where a,b,c>0a,b,c\gt 0, μ,λ∈R\mu ,\lambda \in {\mathbb{R}}, 20b,c\gt 0 and μ∈R\mu \in {\mathbb{R}}, we prove some existence, nonexistence, and asymptotic behavior of the obtained normalized solutions. When μ>0\mu \gt 0:(i) for 2<q<32\lt q\lt 3, we obtain the existence of a local minimizer ground-state solution and a mountain-pass-type solution, (ii) for q=3q=3 and 3<q<43\lt q\lt 4, we obtain the existence of a mountain-pass type ground-state solution respectively, under different assumptions. When μ<0\mu \lt 0 and 2<q<42\lt q\lt 4, we prove the nonexistence result of the aforementioned problem. We also investigate the asymptotic behavior of the normalized ground-state solutions, when μ→0+\mu \to {0}^{+} and b→0+b\to {0}^{+}, respectively.

Keywords

• kirchhoff equation
• normalized solutions
• critical exponent
• combined nonlinearities
• 35j60
• 35j50
• 35j20