Existence and Stability of Normalized Solutions for Nonlocal Double Phase Problems

Abstract

In this paper, we study the following nonlocal double phase problem involving the fractional p-Laplacian

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )_p^\alpha u+\mu \left[ u\right] ^{p-q}_{\beta ,q}(-\Delta )^{\beta }_q u =\lambda \left| u\right| ^{p-2}u+f(x,u)\ \ x\in \Omega ,\\ u=0\ \ \ x\in {\mathbb {R}}^N\setminus \Omega , \end{array}\right. }\end{aligned}$$

where $0<\beta<\alpha<1,\,1<q\le p<{N \over \alpha },\,\mu ,\lambda \in {\mathbb {R}},$ $f\in C(\Omega \times {\mathbb {R}})$. Based on the fractional Gagliardo–Nirenberg inequality, we first give the definition of the $L^p$-mass critical exponent. Then the existence of normalized ground state solutions no matter whether the nonlinearity f is $L^p$-mass subcritical, critical, or supercritical is discussed by using variational methods. In particular, in the supercritical case, we use the eigenvalue of perturbed fractional p-Laplace operator to characterize the conditions for existence of normalized ground states solutions. Finally, we investigate the orbital stability of ground state set of the problem. Our results are new even in the p-Laplacian case.