Non-radial solutions for the critical quasi-linear Hénon equation involving p-Laplacian in \R^N

2025/08/19 by Dai, Wei, Duan, Lixiu, Gui, Changfeng +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.13539

Abstract

In this paper, we investigate the following D^1,p-critical quasi-linear Hénon equation involving p-Laplacian \ \beginaligned amp;-Δ_p u=|x|^αu^{p_\al^*-1}, amp; x∈ \R^N,
amp;ugt;0, amp; x∈ \R^N, \endaligned . where N≥2, 10. By carefully studying the linearized problem and applying the approximation method and bifurcation theory, we prove that, when the parameter \al takes the critical values \al(k):=(p√((N+p-2)²+4(k-1)(p-1)(k+N-1))-p(N+p-2))/(2(p-1)) for k≥2, the above quasi-linear Hénon equation admits non-radial solutions u such that u∼ |x|^-(N-p)/(p-1) and |∇ u|∼ |x|^-(N-1)/(p-1) at ∞. One should note that, α(k)=2(k-1) for k≥2 when p=2. Our results successfully extend the classical work of F. Gladiali, M. Grossi, and S. L. N. Neves in \citeGGN concerning the Laplace operator (i.e., the case p=2) to the more general setting of the nonlinear p-Laplace operator (1

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Cited by

• Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities

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