奇异临界椭圆方程解的存在性

Variational method is a powerful tools in nonlinear analysis theory and could be applied in pure mathematical, applied mathematics (ordinary differential equations, partial differential equations, geometric analysis, measure theory, etc) and even many fields of physics. We study the existence and multiplicity of nontrival solutions for p-Laplace elliptic problems under mixed boundary conditions applying Ekeland’s variational principle, strong maximum principle and Mountain pass Lemma. Furthermore, by applying Pohozaev indentity, we inversitage the nonexistence of solutions under some assumptions of nonlinear terms and domains.. Our results will contribute to a deeper understanding of nonlinear elliptic partial differential equations and providing a solid foundation for the study of the existence and multiplicity of elliptic partial differential equations, and some nonlinear problems; there is a high level cross with non-commutative harmonic analysis; they can communication and deepen the relations between differentdisciplines and branches of mathematics.

变分法是非线性分析理论中一个非常强大的工具, 也可以应用到纯粹数学、应用数学 (常微分方程、偏微分方程、几何分析、测度论等) 甚至物理学的很多领域。利用集中紧性原理，Ekeland 变分原理，强极大值原理，山路引理等变分方法研究一类 p-Laplace椭圆问题在混合Dirichlet-Neumann 边界条件下非平凡解的存在性与多重性。同时，利用Pohozaev 恒等式来研究非线性项和区域满足一定的假设条件下解的不存在性。. 我们的研究将有助于更深入地了解非线性椭圆偏微分方程，为研究椭圆偏微分方程的存在性，多重性和相关非线性问题提供坚实的基础；与非交换调和分析在高层次上实现交叉；沟通与深化不同学科和不同数学分支之间的联系，相互推动和促进发展。

在这个项目里，我们研究了带有 Sobolev-Hardy 临界指数和临界 Hardy 项的奇异椭圆方程在混合 Dirichlet-Neumann 边界条件下正解的存在性，在证明正解的存在性的时候主要用到了集中紧性原理、强极大值原理、山路引理等变分方法。同时，证明了一类带有多重临界 Sobolev-Hardy 指数和多个奇异项的 p-Laplace 椭圆问题在边界区域是严格星形区域时解的不存在性，该结果的证明主要应用了 Pohozaev 恒等式，解的不存在性结果总结了一般拟线性 p-Laplace 椭圆方程解的不存在性。