The Existence of Normalized Solutions for a Nonlocal Problem in ℝ 3
Abstract: In this paper, we study the following fractional Schrödinger equation in ℝ3−Δσu−λu=up−2u, in ℝ3 with σ∈0,1,λ∈ℝ and p∈2+σ,2+4/3σ. By using the constrained variational method, we show the existence of solutions with prescribed L2 norm for this problem.
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Normalized solutions for the fractional Schrödinger equation with combined nonlinearities
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.
Normalized solutions for the fractional Schrödinger equation with combined nonlinearities
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.
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