The anisotropic p-Laplacian version of the Brezis-Nirenberg problem admits positive weak solutions under the same eigenvalue and dimension conditions as the isotropic case.
Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$
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abstract
Given $N\geq 2$, we completely classify the solutions of the anisotropic $N$-Liouville equation $$-\Delta_N^H\,u=e^u \quad\text{in }\mathbb{R}^N,$$ under the finite mass condition $\int_{\mathbb{R}^N} e^u\,dx<+\infty$. Here $\Delta_N^H$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence for $N=2$, we give an affirmative answer to a conjecture made in [G. Wang and C. Xia, J. Differential Equations 252 (2012) 1668--1700].
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Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian
The anisotropic p-Laplacian version of the Brezis-Nirenberg problem admits positive weak solutions under the same eigenvalue and dimension conditions as the isotropic case.