Hugo Tavares
Published 2023-11-15Version 1
In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.
Comments: Review article focused on the author's own work(expanded version of his Habilitation lecture).Draws heavily from: arXiv:2305.02870,arXiv:2211.04839,arXiv:2209.02113, arXiv:2109.14753,arXiv:2106.03661,arXiv:2106.00579,arXiv:1908.11090,arXiv:1807.03082, arXiv:1706.08391, arXiv:1701.05005, arXiv:1508.01783,arXiv:1412.4336,arXiv:1409.5693,arXiv:1405.5549,arXiv:1403.6313,arXiv:1307.3981,arXiv:1201.5206
Subjects: 35B09,
35B33,
35B38,
35B45,
35B65,
35J20,
35J50,
35J61,
35R35,
49Q05,
58J05,
58E05
An interpolation approach to $L^{\infty}$ a priori estimates for elliptic problems with nonlinearity on the boundary
A unified approach to symmetry for semilinear equations associated to the Laplacian in $\mathbb{R}^N$
Multiplicity results for elliptic problems with critical exponential growth
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