Quasilinear nonlocal elliptic problems with prescribed norm in the $L^p$-subcritical and $L^p$-critical growth

Abstract

It is established existence of solution with prescribed $L^p$ norm for the following nonlocal elliptic problem: \begin{equation*} \left\{\begin{array}{cc} \displaystyle (-Δ)^s_p u\ +\ V (x) |u|^{p-2}u\ = λ|u|^{p - 2}u + β\left|u\right|^{q-2}u\ \hbox{in}\ \mathbb{R}^N, \displaystyle \|u\|_p^p = m^p,\ u \in W^{s, p}(\mathbb{R}^N). \end{array}\right. \end{equation*} where $s\in (0,1),sp<N,\beta >0\text{ and }q\in (p,{\overline{p}}_s]$ where ${\overline{p}}_s=p+s{p}^2/N$. The main feature here is to consider $L^p$-subcritical and $L^p$-critical cases. Furthermore, we work with a huge class of potentials $V$ taking into account periodic potentials, asymptotically periodic potentials, and coercive potentials. More precisely, we ensure the existence of a solution of the prescribed norm for the periodic and asymptotically periodic potential $V$ in the $L^p$-subcritical regime. Furthermore, for the $L^p$ critical case, our main problem admits also a solution with a prescribed norm for each $\beta >0$ small enough.