Qualitative and regularity properties of solutions to quasilinear elliptic problems

Abstract

The thesis focuses on the study of the regularity and qualitative properties of solutions to quasilinear elliptic problems. The thesis is divided into two main parts. The first part (chapters 1–5) focuses on problems involving anisotropic functionals, where the main operator is governed by a norm that is generally non-Euclidean, leading to the p-Finsler Laplace operator. The second part (chapters 6–7) examines the regularity properties of solutions to equations involving both the scalar and vectorial Euclidean p-Laplacian operator. • In Chapter 1, we introduce a suitable anisotropic Kelvin-type transformation applied to the Finsler Laplace operator for specific types of anisotropy, particularly the Riemannian case. This transformation leads to a solution that satisfies an anisotropic equation involving the dual norm. • In Chapter 2, we prove Liouville-type theorems for stable solutions (and solutions stable outside a compact set) of quasilinear anisotropic elliptic equations. Specifically, for some general p-Finsler equations, we show that the only stable solutions to suitable nonlinear problems in RN under certain conditions on the dimension of the space and the nonlinearity of the zero-order term of the equation, are identically zero. • In Chapter 3, we establish rigidity results for Finsler p-Laplacian-type equations, including Liouville-type theorems and one-dimensional symmetry results. The proofs are based on a Poincar´e-type inequality and an extension of a formula by Sternberg and Zumbrun, which relates the second derivatives of a weak solution to the principal curvatures of the associated level set, within the anisotropic framework. • In Chapter 4, an asymptotic analysis of solutions to anisotropic doubly critical equations in RN is provided, with particular focus on the behavior of the solutions and their gradients. Specifically, we examine these solutions both near the origin and at infinity, establishing that they exhibit properties analogous to their Euclidean counterparts. • In Chapter 5, we analyze solutions to degenerate anisotropic elliptic equations with a focus on their regularity. Specifically, we derive second-order estimates and establish regularity results for the associated stress field. • Chapter 6 concerns regularity and symmetry properties for the p-Laplacian system. More specifically, we focus on second-order estimates for the stress field. As a consequence of our regularity results, we deduce a weighted Sobolev inequality, which leads to weak comparison principles. Additionally, we apply the moving plane technique to derive symmetry and monotonicity results for the solutions, under suitable assumptions. • Chapter 7 is devoted to the study of solutions to p-Laplace equations as p approaches the semilinear limiting case p = 2. Our focus is on the integrability of the third derivatives of the solutions. As a corollary, the obtained estimates enable us to deduce regularity results for the stress field of the solution. The results presented in this thesis were obtained during my three-year doctoral program. They stem from various scientific collaborations and are included in the following research papers: [15, 16, 77, 78, 91, 92, 128].