Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity

Abstract

In this article, we investigate the following Schrödinger equation: −Δu=h(x)g(u)+λuinRN,∫RN∣u∣2dx=au∈H1(RN),\left\{\begin{array}{ll}-\Delta u=h\left(x)g\left(u)+\lambda u\hspace{1.0em}& \hspace{-0.2em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}\hspace{0.1em}\text{d}\hspace{0.1em}x=a\hspace{1.0em}& u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where N≥3N\ge 3, a>0a\gt 0, and λ∈R\lambda \in {\mathbb{R}} arises as a Lagrange multiplier. Assuming that the sublinear nonlinearity f∈C(RN×R,R)f\in C\left({{\mathbb{R}}}^{N}\times {\mathbb{R}},{\mathbb{R}}) is locally defined for ∣u∣| u| small; by using generalized minimax approach, we prove the existence of infinitely many normalized solutions {(uk,λk)}k∈N{\{\left({u}_{k},{\lambda }_{k})\}}_{k\in {\mathbb{N}}} for the problem, where λk<0{\lambda }_{k}\lt 0 and ∣uk(x)∣→0| {u}_{k}\left(x)| \to 0 as ∣x∣→∞| x| \to \infty . Furthermore, as k→∞k\to \infty , we show that ‖∇uk‖L2(RN)→0{\Vert \nabla {u}_{k}\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{N})}\to 0, ‖uk‖L∞(RN)→0{\Vert {u}_{k}\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{N})}\to 0, and λk→0{\lambda }_{k}\to 0.

Keywords

• schrödinger equation
• infinitely many normalized solutions
• variational method
• local defined nonlinearity
• minimax theorem
• 35j20
• 35j61
• 35b38
• 35a15
• 35d30