Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator

We consider the boundary value problem \begin{document}$\begin{equation*} \begin{cases} -{\rm div}_G(w_1\nabla_G u) = w_2f(u) &\text{ in } \Omega,\\ u=0 &\text{ on } \partial\Omega, \end{cases}\end{equation*}$ \end{document} where \begin{document}$\Omega$\end{document} is a bounded or unbounded \begin{document}$C^1$\end{document} domain of \begin{document}$\mathbb{R}^N$\end{document} , \begin{document}$w_1, w_2 \in L^1_{\rm loc}(\Omega)\setminus\{0\}$\end{document} are nonnegative functions, \begin{document}$f$\end{document} is an increasing function, \begin{document}$\nabla_G$\end{document} and \begin{document}${\rm div}_G$\end{document} are Grushin gradient and Grushin divergence, respectively. We prove some Liouville theorems for stable weak solutions of the problem under suitable assumptions on \begin{document}$\Omega$\end{document} , \begin{document}$w_1$\end{document} , \begin{document}$w_2$\end{document} and \begin{document}$f$\end{document} . We also show the sharpness of our results when \begin{document}$\Omega=\mathbb{R}^N$\end{document} and \begin{document}$f$\end{document} has power or exponential growth.

1. Introduction and main results. Throughout this article we always assume that Ω is a bounded or unbounded C 1 domain of R N , w 1 , w 2 ∈ L 1 loc (Ω) \ {0} are nonnegative functions and f ∈ C 1 ((a, b)) ∩ C 2 ((a, b) \ Z f ) is an increasing function, where −∞ ≤ a < b ≤ +∞ and Z f is the set of zeros of f . Clearly, Z f contains at most one element due to the monotonicity of f . We study the nonexistence of stable weak solutions of the problem −div G (w 1 ∇ G u) = w 2 f (u) in Ω, where the Grushin gradient ∇ G and Grushin divergence div G are defined by for x = (x, y) ∈ R N1 × R N2 = R N and α ≥ 0. We also denote ∆ G u := div G (∇ G u) = ∆ x u+|x| 2α ∆ y u and call ∆ G the Grushin operator, which reduces to the well-known Laplace one when α = 0. For x ∈ R N , we define the Grushin distance from 0 to x as |x| G = |x| 2(α+1) + (α + 1) 2 |y| 2 1 2(α+1) .

PHUONG LE
We may then define Grushin ball B G (x 0 , R) = {x ∈ R N : |x − x 0 | G < R} for x 0 ∈ R N and R > 0. Finally, we denote N α = N 1 + (α + 1)N 2 , which is called the homogeneous dimension associated to the Grushin operator ∆ G .
A weak solution u of problem (1) is said to be stable if The definition of stability is motivated by a phenomenon in physical sciences. A system is called in a stable state if it can recover from perturbations. A small change will not prevent the system from returning to equilibrium. From that intuition, stable solutions are those that make the energy of the system attain a local minimum. In other words, a solution u is stable if the second variation at u of the energy functional is nonnegative. Physical backgrounds and recent developments on stable solutions of semilinear elliptic equations can be found in the interesting monograph [11] by Dupaigne.
Liouville theorem for stable solutions, which concern about nonexistence of this particular type of solutions, have drawn much attention in the last decade. In the pioneering work [12], Farina established a sharp Liouville theorem for stable classical solutions of equation where q > 1. Indeed, he proved that the equation has no nontrivial stable classical solution if and only if q < q c , where is call the Joseph-Lundgren exponent (see [15]). It should be notice that this exponent is larger than the usual Sobolev one q S = N +2 N −2 and condition q < q c is equivalent to N < 2 + 4(q+ √ q(q−1)) q−1 . Later, some of Farina's results were extended to the weighted case in [5,8,23]. In [8], the authors proved the nonexistence of nontrivial stable weak solutions of −∆u = |x| α |u| q−1 u under the restriction that the solutions are locally bounded. This restriction was removed in [23]. Although the work [5] only deals with stable classical solutions, it consider more general class of weights. A typical result in [5] states that if q > 1, w 1 = (|x| 2 + 1) q1/2 , w 2 = (|x| 2 + 1) q2/2 and then there is no positive stable classical solution of −div(w 1 ∇u) = w 2 u q in R N . Since then, these results have been generalized to the weighted quasilinear equation Such generalizations may be found in [6,17,19,20] and references therein. Some attempts to extend Farina's results to elliptic problems with nonlinearity f belonging to a wider class of positive and convex functions were also made in [4,10,16,18]. Another possible generalization corresponds to elliptic problems involving Grushin operator, i.e. problem (1). It is well-known that the operator ∆ G belongs to the wide class of subelliptic operators studied by Franchi et al. in [14] (see also [2,3]). Via Kelvin transform and the method of moving planes, the optimal Liouville type theorem for nonnegative solutions of the problem −∆ G u = u p in R N has been established in [21,24] for the case 1 < p < Nα+2 Nα−2 . More recently, some Liouville type theorems for stable solutions of elliptic problems involving Grushin operator were established in [1,9,22]. In [9], the authors proved the nonexistence of nontrivial stable weak solutions of the equation where N # is explicitly computed. It should be notice that this equation is a special case of (1) where Ω = R N , w 1 = e −w , w 2 = |x| s G e −w and f (u) = |u| q−1 u. The work [22] concerns about nonexistence of stable classical solutions of similar equation with no convection (i.e, w = 0) but with more general subelliptic operator. Moreover, the nonexistence of classical stable solutions of equation −∆ G u = e u in R N was derived in [1] for the case 2 < N α < 10. However, in all of the above works, questions on the sharpness of Liouville theorems are left open, except for the case α = 0. It is due to the difficulty of building explicit stable solutions of such equations when α > 0.
As far as we know, Liouville type theorems for stable weak solutions of (1) has not been fully studied in the literature except for some special cases in [1,9] mentioned above. The purpose of this paper is to establish Liouville results for stable weak solutions of (1) in general domain Ω, with general nonlinearity f and general nonnegative weights w 1 , w 2 .
To state our main result, let us denote by z f the unique zero of f if Z f = ∅. We also denote by B x (x 0 , R) the usual ball in R N1 centered at x 0 with radius R. Similar notation is used for the ball B y (y 0 , R) in R N2 . For a measurable set U ⊂ R N , we denote by |U | its Lebesgue measure. We also use the convention that U 1 v dx = +∞ if v is a nonnegative function which is equal to zero in a subset of U having positive Lebesgue measure. Finally, for c ∈ R we define sign(c) = 1 if c ≥ 0 and sign(c) = −1 otherwise.
and there exists β Assume in addition that Although this paper is motivated by the idea of Farina [12] and related works, it should be mentioned that the use of available techniques in our case is not straightforward and many nontrivial additional ideas are introduced to overcome the difficulties caused by the generality of our statements.
• We consider a large class of weights w 1 , w 2 and nonlinearity f and we neither assume positivity nor convexity of f . We also consider the equation in a proper domain Ω ⊂ R N with Dirichlet boundary condition. This is different from other works which deal with general nonlinearity (see [4,10,16]) and therefore our results require very delicate analysis. • The fact that weak solutions are not locally bounded also leads to another difficulty. The usual cut-off functions as used in [9,23] do not work with general nonlinearity f . To overcome this difficulty, we use new cut-off functions inspired by [17,19,20] and show that our key estimates are still valid for these functions. • We also construct some examples to show the sharpness of our Liouville theorems. To prove the stability of constructed solutions, we use a version of Hardy inequalities related to Grushin type operators established in [7]. As a consequence of Theorem 1.2, we have the following Liouville result which is valid for problems in a bounded domain in any dimension.
For the next result, we say that Proposition 2. Assume that f (0) = 0 or Ω = R N , and one of the following two conditions occurs We also assume in addition that for some ε > 0, where (6) is also satisfied in the case that Ω has finite Lebesgue measure and N # > 0. If we focus our attention to the class of bounded stable weak solutions, then we are able to establish a Liouville theorem in R N without any restriction on the homogeneous dimension N α . Theorem 1.3. Assume that f is positive and a = −∞, or f is negative and b = +∞. Assume in addition that there exists γ > 0 such that γf (t) 2 ≤ f (t)f (t) for all t ∈ (a, b) and w 1 , w 2 satisfy assumption (W'). Then the problem has no bounded stable weak solution.
As an application, we may derive the following Liouville theorems for problem (1) with Lane-Emden, negative exponent, exponential or singular nonlinearity.
Proposition 3. Assume that q > 1 and w 1 , w 2 satisfy (W) with Γ = γ = q−1 q . Then zero is the unique stable weak solution of problem In particular, the conclusion holds in one of the following cases (i) Ω is bounded, (ii) Ω has finite Lebesgue measure and N # > 0, Proposition 4. Assume that q < 0 and w 1 , w 2 satisfy (W) with Γ = γ = q−1 q . Then equation div G (w 1 ∇ G u) = w 2 u q in R N has no positive stable weak solution in homogeneous dimension Moreover, the equation has no bounded positive stable weak solution in any dimension if w 1 , w 2 satisfy (W').
Proposition 6. Assume that w 1 , w 2 satisfy (W') and f : (0, +∞) → R has one of the following forms has no bounded positive stable weak solution.

Remark 1.
The assumption on N α < N # in Proposition 3 can be rewritten as .
Note that q c (N, 0, 0) is the Joseph-Lundgren exponent as mentioned in the introduction of this paper. Proposition 3 recovers some known results in [5,8,12,23] when α = 0. A result similar to Proposition 3 has been established in [9] for Ω = R N under more restrictions on w 1 and w 2 . The main novelty of Proposition 3 is therefore the conclusion in the case Ω = R N .
We also point out that the first part of Proposition 5 (i.e., nonexistence of stable weak solutions) has been obtained in [1, Corollary 1.3] under additional assumptions that w 1 = w 2 = 1 and u ∈ C 2 (R N ).
Remark 2. Proposition 4 and 6 are completely new to the best of our knowledge. To prove Proposition 6, it suffice to show that f (t)f (t) Let us emphasize that Proposition 2 is sharp in the sense that problem (1) may have nontrivial stable weak solutions if (6) is not satisfied for any ε > 0, which is the case if Ω = R N and N α ≥ N # . Therefore, we may call N # a critical homogeneous dimension, which divides the range of instability of problem (1). Indeed, if Ω = R N and f has power or exponential growth, we are able to give some examples of such stable weak solutions.
. We define f (u) = sign(q)u q , Then f , w 1 and w 2 satisfy (F1) and (W) with Γ = γ = q−1 q . Furthermore, if then U (x) = |M | 1 q−1 |x| Then f , w 1 and w 2 satisfy (F1) and (W) with Γ = γ = 1. Furthermore, if In what follows, we denote by C a generic constant whose concrete values may change from line to line or even in the same line. If this constant depends on an arbitrary small number ε, then we may denote it by C ε . For the sake of simplicity, we also denote by v the integral Ω v dx.
We prove our main results, namely Theorem 1.2, 1.3 and Proposition 2, in the next section. Section 3 is devoted to the sharpness of our Liouville theorems where we prove Proposition 7 and 8.
2. Liouville type theorems for stable weak solutions. The key estimate in the proof of our main results is (17). In order to obtain it, we firstly derive an integral estimate which is valid for all f ∈ C 1 ((a, b)).
Proof of Proposition 2. We have Therefore, if f satisfies (F1), then f /|f | Γ is non-decreasing in (a, c) and nonincreasing in (c, b), where c is defined as in proof of Theorem 1.2. Hence, f (t) ≥ C|f (t)| Γ and (4) is satisfied. Otherwise, if f satisfies (F2), then f /|f | Γ is nonincreasing in (a, c) and non-decreasing in (c, b). Hence also in this case, f (t) ≥ C|f (t)| Γ and (4) is satisfied.
Proof of Theorem 1.3. We point out that f (t) > 0 for all t ∈ (a, b). Indeed, assume that f (t 0 ) = 0 for some t 0 ∈ (a, b). If f is positive, then f (t) ≥ 0 for all t ∈ (−∞, b). Therefore, f (t) ≤ f (t 0 ) = 0 for all t ∈ (−∞, t 0 ]. But this contradicts to the fact that f is increasing in t ∈ (−∞, t 0 ]. Similar argument can be carried out in the case that f is negative. By contradiction, assume that (1) has a bounded stable weak solution u. Then we may restrict f into interval − u L ∞ (R N ) − 1, u L ∞ (R N ) + 1 ∩ (a, b) and find out that f satisfies (F2) in its new domain for any Γ ∈ (0, γ]. If we choose Γ sufficiently close to 0 such that 2(1 + √ γ)(q 2 − q 1 + 2) we reach a contradiction by applying Proposition 2.
3. The sharpness of the critical homogeneous dimension. To show the sharpness of the critical homogeneous dimension N # in Proposition 2, we utilize the following Hardy type inequality involving Grushin gradient.
Proposition 10 (Hardy type inequality [7]). Let r, s ∈ R be such that N α +2 > r−s and N 1 > 2α − s. Then for every ϕ ∈ C 1 c (R N ), we have The above inequality is actually a special case of [7, Theorem 3.1], but it is sufficient for our purpose. We also need some basic formulas whose proofs are provided for readers' convenience.