Normalized solutions for the fractional Schrödinger equation with combined nonlinearities

Abstract

In this paper, we study the normalized solutions for the following fractional Schrödinger equation with combined nonlinearities { ( - Δ ) s ⁢ u = λ ⁢ u + μ ⁢ | u | q - 2 ⁢ u + | u | p - 2 ⁢ u in ⁢ ℝ N , ∫ ℝ N u 2 ⁢ 𝑑 x = a 2 , \displaystyle\left\{\begin{aligned} \displaystyle{}(-\Delta)^{s}u&% \displaystyle=\lambda u+\mu\lvert u\rvert^{q-2}u+\lvert u\rvert^{p-2}u&&% \displaystyle\phantom{}\text{in }\mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}}u^{2}\,dx&\displaystyle=a^{2},\end{aligned}\right. where 0 < s < 1 {0<s<1} , N > 2 ⁢ s {N>2s} , 2 < q < p = 2 s * = 2 ⁢ N N - 2 ⁢ s {2<q<p=2_{s}^{*}=\frac{2N}{N-2s}} , a , μ > 0 {a,\mu>0} and λ ∈ ℝ {\lambda\in\mathbb{R}} is a Lagrange multiplier. Since the existence results for p < 2 s * {p<2_{s}^{*}} have been proved, using an approximation method, that is, let p → 2 s * {p\rightarrow 2_{s}^{*}} , we obtain several existence results. Moreover, we analyze the asymptotic behavior of solutions as μ → 0 {\mu\rightarrow 0} and μ goes to its upper bound.