A topological approach to an elliptic problem

Technical Summary: A Topological Approach to an Elliptic Problem

Problem Statement
The paper investigates a boundary value problem involving a quasilinear elliptic operator driven by a $p$-Laplacian, a potential well, and a combination of singular and critical nonlinearities. The specific equation under study is:
$$\begin{cases} -\Delta_p u + \mu V(x)|u|^{p-2}u = \beta|u|^{p^*-2}u + \lambda|u|^{-\gamma-1}u & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases}$$
where $\Omega \subset \mathbb{R}^N$ ($N \ge 3$) is a bounded domain with a smooth boundary, $1 < p < N$, and $p^* = \frac{Np}{N-p}$ is the critical Sobolev exponent. The parameters satisfy $\mu, \beta, \lambda > 0$ and $0 < \gamma < 1$. The potential $V \in L^\infty(\Omega)$ is non-negative, with the interior of its zero set, $\tilde{\Omega} = \text{int}(V^{-1}(0))$, being non-empty.

The problem presents two primary analytical challenges:

1. Critical Growth: The term $|u|^{p^*-2}u$ leads to a lack of compactness in Sobolev embeddings, a difficulty traditionally addressed via the concentration-compactness principle.
2. Singular Nonlinearity: The term $|u|^{-\gamma-1}u$ introduces degeneracy in the energy functional near zero, rendering the functional ill-defined on the entire Sobolev space $W^{1,p}_0(\Omega)$.

Methodology
The authors adopt a topological approach distinct from the standard variational methods (such as Nehari manifold decomposition) often used for concave-convex problems. The core methodology relies on Abstract Critical Point Theory utilizing cohomological indices and genus theory.

1. Function Space: The problem is analyzed in the weighted space $W_\mu$, defined as the closure of $C_0^\infty(\Omega)$ under the norm $\|u\|_\mu^p = \int_\Omega (|\nabla u|^p + \mu V(x)|u|^p) dx$. This space is a reflexive Banach space.
2. Critical Point Theorems:
   • Perera's Theorem: The authors utilize a theorem by Perera involving the cohomological index to establish the existence of multiple solutions. This theorem allows for the identification of critical values $c_k^*$ associated with symmetric sets of a specific index.
   • Rabinowitz's Symmetric Mountain Pass Theorem: To prove the existence of infinitely many solutions, the paper employs Rabinowitz's theorem, which requires the functional to be even, satisfy the Palais-Smale condition, and exhibit specific geometric behaviors on finite-dimensional subspaces.
3. Regularity Analysis: To handle the singularity, the authors first establish that weak solutions are essentially bounded ($L^\infty$). This is achieved through:
   • A weak Harnack-type inequality derived via Moser iteration for positive supersolutions, providing a positive lower bound on compact subsets.
   • A De Giorgi-type iteration to prove $L^\infty$ boundedness.
   • Once boundedness is established, the singular term becomes locally bounded, allowing the application of classical regularity results by Tolksdorf and Lieberman to upgrade the solution to $C^{1,\alpha}_{loc}(\Omega)$.

Key Results
The paper establishes the following main results regarding the existence and multiplicity of weak solutions:

1. Multiplicity for Large $\beta$: For any $\beta \ge \beta^*$, the problem admits $m$ distinct pairs of weak solutions with positive critical values.
   • Specifically, for $k \ge 2$, there exist infinitely many critical points with critical energy $c_k^*$.
   • For $k=1$, there exist exactly two critical points, $u$ and $-u$.
2. Infinite Multiplicity: Irrespective of the value of the parameter $\beta > 0$, the problem admits infinitely many solutions.
3. Regularity: Every positive weak solution $u$ belongs to $C^{1,\alpha}_{loc}(\Omega)$ for some $\alpha \in (0,1)$. This is proven by showing that the nonlinear term $f(x) = \beta u^{p^*-1} + \lambda u^{-\gamma} - \mu V(x)u^{p-1}$ belongs to $L^\infty_{loc}(\Omega)$, effectively treating the equation locally as a $p$-Laplace equation with a bounded right-hand side.
4. Independence of Solution Sets: The authors note that the sets of solutions obtained via Rabinowitz's theorem (using finite-dimensional compact symmetric sets) and Perera's theorem (using sets with cohomological index) may differ, even if they share critical energy levels, due to the existence of multiple solutions at specific energy levels.

Significance and Claims
The paper claims to provide a topological framework for solving elliptic problems that combine singular and critical nonlinearities, an area where standard variational techniques often face difficulties due to the lack of compactness and the undefined nature of the energy functional at zero.

• Methodological Contribution: By shifting from Nehari manifold analysis to cohomological index theory, the authors demonstrate a robust method for handling the competition between concave (singular) and convex (critical) effects.
• Regularity: The work explicitly bridges the gap between the existence of weak solutions and their regularity in the presence of singular terms, proving that the singularity does not prevent solutions from being $C^{1,\alpha}$ locally.
• Limiting Behavior: The abstract mentions a limiting case where a parameter blows up to 8, yielding solutions to a different problem where the potential well effect becomes negligible, though the detailed analysis of this specific limit is not the primary focus of the provided text's proofs.

The authors position their work as an extension of previous studies on semilinear and quasilinear problems (citing works by Ambrosetti-Brezis-Cerami, Giacomoni et al., and others) by specifically addressing the interplay of singular and critical terms through topological tools.