Normalized solutions to a quasilinear equation involving critical Sobolev exponent

TL;DR

This paper investigates the existence and regularity of normalized solutions to a complex quasilinear elliptic equation involving critical Sobolev exponents, fractional operators, and nonlocal terms, advancing understanding of such equations in mathematical physics.

Contribution

It introduces new existence and regularity results for normalized solutions to a quasilinear Choquard equation with critical Sobolev exponent and mixed diffusion operators.

Findings

Existence of normalized solutions under specified conditions.

Regularity results for solutions in the fractional and nonlocal setting.

Extension of known results to equations with critical exponents and mixed operators.

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.