Topics in elliptic problems: from semilinear equations to shape optimization

Abstract

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.

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

References in corpus (9)

• Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation
• Sign-changing solutions to the slightly supercritical Lane-Emden system with Neumann boundary conditions
• Asymptotic behaviour of a semilinear elliptic system with a large exponent
• An asymptotic relationship between Lane-Emden systems and the 1-bilaplacian equation
• Spectral partition problems with volume and inclusion constraints
• Optimal uniform bounds for competing variational elliptic systems with variable coefficients
• Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two
• Existence of solutions to a slightly supercritical pure Neumann problem
• Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions