A priori estimates of stable and finite Morse index solutions to elliptic equations that arise in Physics

Abstract

The study of qualitative properties, such as existence, uniqueness/multiplicity, and regularity of solutions to the nonlinear Poisson equation −Δu = f(x, u) subject to Dirichlet boundary conditions on a given domain (NPDP), constitutes a classical problem with a vast body of literature. It continues to be an active area of research at the international level, alongside its various generalizations to nonlinear, nonlocal operators and alternative boundary conditions.

Within the current landscape of the mathematical research community, particularly in the field of partial differential equations, the analysis of regularity properties of solutions has garnered significant attention, both for its theoretical interest and the inherent difficulties it presents. Over the last two decades, the notion of solution stability has emerged as a powerful tool to guarantee regularity in elliptic equations. A solution is said to be stable if the second variation of the associated energy functional is nonnegative definite. Alternatively, the stability of a solution can be quantified via its Morse index, which, roughly speaking, is defined as the dimension of the largest subspace of test functions on which the second variation is negative definite.

This theoretical framework has yielded profound results. A notable example is the work of Cabré, Figalli, Ros-Oton, and Serra [Cabré et al., 2020], who simultaneously established the Hölder continuity of stable solutions to (NPDP) in the optimal range of dimensions N ≤ 9, as well as definitively solved a conjecture by Brezis and Vázquez concerning the regularity of the extremal solution. These results have not yet been generalized to solutions with finite Morse index (as opposed to infinite Morse index, for which no such regularity is expected). Nevertheless, there exist partial results: for example, [Bahri and Lions, 1992] for subcritical nonlinearities, and [Figalli and Zhang, 2024] for supercritical ones. These works manage to obtain a priori bounds for the solutions in terms of constants that depend only on the dimension and the Morse index.

A substantial portion of this thesis is situated within this research direction. In Chapter 3, we construct a counterexample to what we refer to as the “Extended Brezis–Vázquez Conjecture,” namely, the conjecture regarding solutions with finite Morse index. In particular, we demonstrate that bounding the radial Morse index does not, contrary to possible expectations, suffice to preclude the occurrence of singular solutions. We construct a sequence of solutions to problems of the form (NPDP) whose radial Morse index is necessarily equal to one, and for which the quotient ||·||ₚ / ||·||_q diverges for every pair 1 ≤ q < p ≤ ∞ with p > N/(N − 2) (the constraint p > N/(N − 2) appears to be unexpected and possibly of a technical nature), in dimensions 3 ≤ N ≤ 9.

In Chapter 4, we establish regularity results for radial solutions of a non-autonomous version of the equation, namely, the Hardy–Hénon equation: −Δu = |x|^α f(u), determining the optimal range of dimensions 2 ≤ N ≤ 10 + 4α, assuming α > −2. The case α ≤ −2 is much more pathological and there are even some Liouville results for general weak solutions (see [Dancer et al., 2011]).

In Chapter 5, we also address the problem of existence and multiplicity of solutions to the equation −Δu = g(u) − h(x)f(u), where h ∈ L∞(Ω) is a non-negative, nontrivial function satisfying h > 0 a.e. in Ω \ Ω₀, with Ω₀ = interior({x ∈ Ω / h(x) = 0}), and |Ω₀| > 0. Due to the vanishing property of h, this class of problems has been less addressed in the literature. In this work, we prove the existence of both a positive and a negative solution, as well as up to four distinct nontrivial solutions, under the assumptions that g is asymptotically linear and that the nonlinearity f satisfies a suitable set of conditions. A crucial point is the relative position of the asymptotic slope λ = g′(0) with respect to the spectrum of the self-adjoint operator associated to a quadratic form defined on the subdomain Ω₀, a key aspect already highlighted in [Alama and Li, 1992]. The main tools employed include the sub- and supersolution method in its integral formulation to obtain two sign-constant solutions, the Mountain Pass Lemma to find a third, and the critical groups associated to all of them to identify and distinguish a fourth solution (see Theorem 1.19 or [Chang, 1993, Theorem 3.5]).

Finally, Chapter 6 outlines several avenues for future research. The results obtained herein admit various generalizations. In particular, whether the counterexample in Chapter 3 genuinely invalidates the extended version of the Brezis–Vázquez conjecture for arbitrary finite Morse index remains an open question. Additionally, the potential applicability of the techniques developed in [Cabré et al., 2020] to the study of the regularity of stable solutions of the non-autonomous Hardy–Hénon equation on convex domains is a promising direction worth further exploration.

The notions of stability explored in the present thesis may also find applications in the study of stability phenomena in theoretical physics and field theory. Issues such as the convergence of the perturbative series and the discrepancy between linear and higher-order approximations naturally arise in this context. At present, any physically viable theory must withstand stringent stability criteria. Nonetheless, a lack of precise methods for ruling out all physically undesirable instabilities persists. Significant progress would be achieved by formulating rigorous definitions and theorems capable of detecting and ruling out such instabilities—such as strong coupling. A representative case is that of Einstein cubic gravity, as discussed in [Beltrán Jiménez and Jiménez-Cano, 2021].