Multiple normalized solutions for fractional elliptic problems

Abstract

In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional p-Laplace problem: { ( - Δ ) p s ⁢ v + 𝒱 ⁢ ( ξ ⁢ x ) ⁢ | v | p - 2 ⁢ v = λ ⁢ | v | p - 2 ⁢ v + f ⁢ ( v ) in ⁢ ℝ N , ∫ ℝ N | v | p ⁢ 𝑑 x = a p , \left\{\begin{aligned} \displaystyle{}(-\Delta)_{p}^{s}v+\mathcal{V}(\xi x)% \lvert v\rvert^{p-2}v&\displaystyle=\lambda\lvert v\rvert^{p-2}v+f(v)\quad% \text{in }\mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}}\lvert v\rvert^{p}\,dx&\displaystyle=a^{p},% \end{aligned}\right. where a , ξ > 0 {a,\xi>0} , p ≥ 2 {p\geq 2} , λ ∈ ℝ {\lambda\in\mathbb{R}} is an unknown parameter that appears as a Lagrange multiplier, 𝒱 : ℝ N → [ 0 , ∞ ) {\mathcal{V}:\mathbb{R}^{N}\to[0,\infty)} is a continuous function, and f is a continuous function with L p {L^{p}} -subcritical growth. We prove that there exists the multiplicity of solutions by using the Lusternik–Schnirelmann category. Next, we show that the number of normalized solutions is at least the number of global minimum points of 𝒱 {\mathcal{V}} , as ξ is small enough via Ekeland’s variational principle.