发表机构
Universidade Federal da Paraíba; Florida Institute of Technology（帕拉伊巴联邦大学; 佛罗里达理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究混合阶拟线性椭圆方程，建立紧加权嵌入与变分框架，证明解的存在性与多重性，并利用势估计和自举方法获得局部正则性及Pohozaev型恒等式。
AI 中文摘要
我们研究了一类由q-双调和算子和p-拉普拉斯算子驱动的混合阶拟线性椭圆方程，即Δ_q^2u-Δ_pu = λ|u|^{r_σ-2}u/|x|^σ + |u|^{p^*-2}u + |u|^{q^{**}-2}u，其中1<q<N/2，1<p<q^*，p≠q，q≠p^*，加权指数由方程的自然尺度决定。我们建立了与该混合尺度相容的紧加权嵌入，并发展了相应的变分框架。作为应用，我们在参数的适当假设下证明了非平凡解的存在性和多重性。我们还针对更一般的方程Δ_q^2u-Δ_pu=f(x,u)建立了局部正则性结果，其中f是具有局部临界增长的Carathéodory函数。通过结合非线性势估计、正则性提升论证以及相关二阶系统的有限自举方法，我们获得了u和非线性通量|Δu|^{q-2}Δu的更高局部正则性。这一正则性结果也具有独立意义，使我们能够推导出所考虑方程的Pohozaev型恒等式。
英文摘要
We investigate a class of mixed-order quasilinear elliptic equations in \(\mathbb R^N\) driven by the \(q\)-biharmonic and \(p\)-Laplacian operators, \[ Δ_q^2u-Δ_pu = λ\frac{|u|^{r_σ-2}u}{|x|^σ} +|u|^{p^*-2}u +|u|^{q^{**}-2}u, \] where \(1<q<N/2\), \(1<p<q^*\), \(p\ne q\), \(q\ne p^*\), and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation $Δ_q^2u-Δ_pu=f(x,u),$ where \(f\) is a Carathéodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both \(u\) and the nonlinear flux \(|Δu|^{q-2}Δu\). This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.
