Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4

Abstract

Abstract In this article, we study the following Kirchhoff equation with combined nonlinearities: − a + b ∫ R 4 ∣ ∇ u ∣ 2 d x Δ u + λ u = μ ∣ u ∣ q − 2 u + ∣ u ∣ 2 u , in R 4 , ∫ R 4 ∣ u ∣ 2 d x = c 2 , \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 > 0 a,b,c\gt 0 , μ , λ ∈ R \mu ,\lambda \in {\mathbb{R}} , 2 < q < 4 2\lt q\lt 4 . Under different assumptions on b , c > 0 b,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 < 3 2\lt q\lt 3 , we obtain the existence of a local minimizer ground-state solution and a mountain-pass-type solution, (ii) for q = 3 q=3 and 3 < q < 4 3\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 < 4 2\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.