Multiplicity of solutions to a class of degenerate elliptic equations in both sub-critical and critical cases

Technical Summary: Multiplicity of Solutions to a Class of Degenerate Elliptic Equations in Both Sub-critical and Critical Cases

Problem Statement
This paper investigates the multiplicity of non-trivial, non-negative weak solutions for a semilinear degenerate elliptic equation involving the Grushin Laplacian operator, $\Delta_\lambda$. The problem is formulated on a smooth, bounded domain $\Omega \subset \mathbb{R}^N$ (where $N=n+m$) as:
$$\begin{cases} -\Delta_\lambda u = \mu g(z)|u|^{r-1}u + h(z)|u|^{s-1}u & \text{in } \Omega, \\ u \in H^1_{0,\lambda}(\Omega), \end{cases}$$
where $\Delta_\lambda = \Delta_x + |x|^{2\lambda}\Delta_y$ is the Grushin operator, $z=(x,y)$, $\lambda > 0$, and $\mu > 0$ is a parameter. The exponents satisfy $0 \le r < 1 < s < 2^*_\lambda - 1$ in the sub-critical case, and $s = 2^*_\lambda - 1$ in the critical case, where $2^*_\lambda = \frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q = n + (1+\lambda)m$. The weight functions $g$ and $h$ are allowed to change sign, subject to specific integrability and local boundedness conditions.

The study addresses a gap in the existing literature regarding the Grushin operator. While existence results for Brezis-Nirenberg type problems (where $r \ge 1$) have been established, the case $0 \le r < 1$ (concave-convex nonlinearity) had not been studied for this operator, even in the sub-critical regime. Furthermore, multiplicity results for convex-concave problems involving the Grushin operator remain less understood compared to the standard Laplacian.

Methodology
The authors employ variational methods, specifically analyzing the critical points of the associated energy functional $I_\mu: H^1_{0,\lambda}(\Omega) \to \mathbb{R}$. The core of the analysis relies on the Nehari manifold technique.

1. Variational Framework: The functional $I_\mu$ is defined as:
   $$I_\mu(u) = \frac{1}{2}\int_\Omega |\nabla_\lambda u|^2 dz - \frac{\mu}{r+1}\int_\Omega g|u|^{r+1} dz - \frac{1}{s+1}\int_\Omega h|u|^{s+1} dz.$$
   The Nehari manifold $\mathcal{M}_\mu$ is defined as the set of non-zero functions where the derivative of the functional in the direction of the function vanishes ($\langle I'_\mu(u), u \rangle = 0$).

2. Fibering Maps: The authors analyze the fibering map $F(t) = I_\mu(tu)$ to decompose the Nehari manifold into three disjoint sets: $\mathcal{M}^+_\mu$, $\mathcal{M}^-_\mu$, and $\mathcal{M}^0_\mu$, based on the sign of the second derivative of the fibering map.

   • For small $\mu$, it is shown that $\mathcal{M}^0_\mu$ is empty.
   • The manifold is split such that $\mathcal{M}^+_\mu$ and $\mathcal{M}^-_\mu$ are non-empty and contain local minimizers of the functional.

3. Sub-critical Case ($s < 2^*_\lambda - 1$):

   • The authors prove that for sufficiently small $\mu$, the functional attains its infimum on both $\mathcal{M}^+_\mu$ and $\mathcal{M}^-_\mu$.
   • Using Ekeland's variational principle, they construct minimizing sequences and prove they converge strongly to critical points of $I_\mu$, yielding two distinct non-negative solutions.
   • The strict positivity of solutions is established under additional conditions ($n=1, \lambda \ge 1$, $g,h \ge 0$) by citing a maximum principle result from previous literature.

4. Critical Case ($s = 2^*_\lambda - 1$):

   • The presence of the critical exponent introduces a lack of compactness (Palais-Smale condition failure).
   • To overcome this, the authors adopt methods from Brezis and Nirenberg. They establish a threshold level $c^*$ (related to the best Sobolev constant $S_\lambda$) below which the Palais-Smale condition holds.
   • First Solution: Obtained via Ekeland's principle on a small ball around the origin, yielding a solution with negative energy.
   • Second Solution: Obtained via the Mountain Pass Theorem. The authors construct a specific path using a family of test functions $w_\epsilon$ (derived from extremal functions of the Sobolev inequality) to show that the mountain pass level lies strictly below the critical threshold $c^*$. This ensures the existence of a second critical point.

Key Contributions and Results
The paper establishes the following main theorems:

• Theorem 1.1 (Sub-critical Multiplicity): Assuming $0 \le r < 1 < s < 2^*_\lambda - 1$ and appropriate conditions on weights $g$ and $h$, there exists $\mu^* > 0$ such that for every $\mu \in (0, \mu^*)$, the problem admits at least two non-trivial, non-negative weak solutions. If $n=1, \lambda \ge 1$ and weights are non-negative, these solutions are strictly positive.
• Theorem 1.2 (Critical Multiplicity): In the critical case where $s = 2^*_\lambda - 1$ and $g \ge 0, h \equiv 1$, there exists $\mu^* > 0$ such that for every $\mu \in (0, \mu^*)$, the problem admits at least two non-trivial, non-negative weak solutions. Under the same conditions on dimension and weights, the solutions are strictly positive.

Significance and Scope
The paper's significance lies in extending the classical Ambrosetti-Brezis-Cerami results (originally for the standard Laplacian) to the degenerate Grushin setting with sign-changing weights. It specifically fills the void regarding the concave-convex case ($0 \le r < 1$) for the Grushin operator.

The authors explicitly note that due to the degeneracy of the operator on the set $\{x=0\}$, a suitable maximum principle is generally unavailable for $n \ge 2$, restricting their focus to non-negative solutions. The work relies heavily on the structural properties of the Grushin operator, including its hypoellipticity and associated Sobolev embeddings, to handle the lack of uniform ellipticity. The results are purely analytical, relying on variational techniques and asymptotic analysis of extremal functions, without proposing new experimental applications or future directions beyond the mathematical scope of the established theorems.