Normalized solutions to a quasilinear equation involving critical Sobolev exponent

Abstract

In this paper we study the existence and regularity results of normalized solutions to the following quasilinear elliptic Choquard equation with critical Sobolev exponent and mixed diffusion type operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u+(-\Delta_p)^su & = & \lambda |u|^{p-2}u +|u|^{p^*-2}u+ \mu(I_{\alpha}*|u|^q)|u|^{q-2}u\;\;\text{in } \mathbb{R}^N, \int_{\mathbb{R}^N}|u|^pdx & = & \tau, \end{array} \end{equation*} where $N\geq 3$, $\tau>0$, $\frac{p}{2}(\frac{N+\alpha}{N})<q<\frac{p}{2}(\frac{N+\alpha}{N-p})$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,N)$, $\mu>0$ is a parameter, $(-\Delta_p)^s$ is the fractional p-laplacian operator, $p^*=\frac{Np}{N-p}$ is the critical Sobolev exponent and $\lambda$ appears as a Lagrange multiplier.