Normalized groundstates for mixed $(p,2)$-Laplacian equations in $\mathbb R^2$ with exponential critical growth

TL;DR

This paper establishes the existence of normalized groundstates for mixed $(p,2)$-Laplacian equations in \\mathbb{R}^2$ with exponential critical growth, overcoming regularity and compactness challenges via a refined Moser iteration and variational methods.

Contribution

It introduces a novel approach combining a refined Moser iteration and constrained minimization to prove groundstates existence without sign restrictions on the Lagrange multiplier.

Findings

Proves existence of normalized groundstates for all positive masses.

Develops a refined Moser iteration technique for exponential critical growth.

Establishes the Pohozaev identity under minimal regularity assumptions.

Abstract

We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*} \begin{cases} -\Delta_p u-\Delta u+\lambda u=f(u) & \text{in } \mathbb{R}^2, \displaystyle \int_{\mathbb{R}^2}|u|^2\,\mathrm{d}x=m, u\in H^1(\mathbb{R}^2)\cap D^{1,p}(\mathbb{R}^2), \end{cases} \end{align*} where $\Delta_p$ denotes the $p$-Laplacian with $1<p<2$, $\lambda\in\mathbb{R}$ represents a Lagrange multiplier and the nonlinerity $f$ exhibits exponential critical growth. Compared to the single-Laplacian case, the lack of regularity here precludes the Pohozaev identity, and the exponential critical growth severely compromises the restoration of compactness. To address these issues, we introduce a refined Moser iteration technique adapted to exponential critical growth, which establishes the Pohozaev identity for weak solutions under the mere assumption of $C_{\mathrm{loc}}^{1,\alpha}$-regularity. By combining constrained minimization on the Pohozaev manifold within a closed $L^2$-ball with a minimax characterization, we prove the existence of normalized groundstates for any prescribed mass $m>0$. Notably, our approach works independently of the sign of the Lagrange multiplier $\lambda$, thereby surmounting the fundamental barrier in recovering compactness for mixed Laplacian problems.