Liouville type theorem for stable solutions to elliptic equations involving the grushin operator

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2019-0026 Natural Science, 2019, Volume 64, Issue 6, pp. 12-22 This paper is available online at LIOUVILLE TYPE THEOREM FOR STABLE SOLUTIONS TO ELLIPTIC EQUATIONS INVOLVING THE GRUSHIN OPERATOR Nguyen Thi Quynh Faculty of Fundamental Science, Hanoi University of Industry Abstract. We study a Liouville type theorem for stable solutions of the following semilinear equation involving Grushin operators −(∆xu + a2|x|2α∆yu) = |u|p−1u, (x, y) ∈ RN = RN1×RN2 ,where p > 1, α > 0 and a 6= 0. Basing on the technique of Farina [1], we establish the nonexistence of nontrivial stable solutions under the range p < pc(Nα) where Nα = N1+(1+α)N2, and pc(Nα) is a certain (explicitly given) positive constant depending on Nα. Keywords: Liouville type theorem, stable solution, degenerate elliptic equations, Grushin operators. 1. Introduction In this paper, we study the semilinear degenerate partial differential equation of the form −Gαu = |u|p−1u (1.1) where Gα = ∆x + a2|x|2α∆y is the Grushin operator, ∆x and ∆y are Laplace operators with respect to x ∈ RN1 and y ∈ RN2 . Here we always assume that a 6= 0 , α > 0, p > 1 and N1, N2 ≥ 1. Recall that Gα is elliptic for |x| 6= 0 and degenerates on the manifold {0} × RN2 . This operator belongs to the wide class of subelliptic operators studied by Franchi et al. in [2]. In the special case α = 1, problem (1.1) is close related to the Heisenberg Laplacian equation ∆Hu = f(u) in Hn = Cn × R, where ∆H is the Heisenberg Laplacian (see e.g., [3, 4]). Problem (1.1) has recently attracted much attention in variety of mathematics directions. The most interesting questions are about the existence and non-existence [5]; the multiplicity of solutions [6]; the symmetry properties [7]; the asymptotic behaviour [8]; the regularity estimates [9-11]. Received April 19, 2019. Revised June 19, 2019. Accepted June 26, 2019. Contact Nguyen Thi Quynh, e-mail address: nguyen.quynh@haui.edu.vn 12 Liouville type theorem for stable solutions to elliptic equations involving the Grushin operator Let us recall that the Liouville-type theorem is the nonexistence of nontrivial solution of problem (1.1) in the whole space RN = RN1 × RN2 . In recent years, the Liouville property has emerged as one of the most powerful tools in the study of qualitative properties for nonlinear PDEs. It turns out that one can obtain from Liouville type theorems a variety of results such as universal, pointwise, a priori estimate; universal and singularity estimates; decay estimates, etc., see [12] and references therein. In addition, Liouville type results combined with degree type arguments are useful to obtain the existence of solutions of semilinear boundary value problems in bounded domains (see [13]). In what follows, we make a short review on the recent developments of Liouville-type property for the problem (1.1). In the class of nonnegative solutions, it has been recently proved by Monticelli [14] for nonnegative classical solutions, and by Yu [15] for nonnegative weak solutions. The optimal condition on the range of the exponent is p < Nα+2 Nα−2 , where Nα := N1 + (1 + α)N2 is called the homogeneous dimension. The main tool in [14, 15] is the Kelvin transform combined with technique of moving planes. Before that, Dolcetta and Cutrı` [16] established the Liouville-type theorem for nonnegative super-solutions under the condition p ≤ Nα Nα−2 (see also [17]). Further results on Liouville type theorem on manifold was established in [18]. In the the class of sign-changing solutions, the Liouville-type theorem is still open, even in the special case of Laplace operator with α = 0. However, in a special class of solutions – the so-called stable solutions, the Liouville type result for α = 0 was completely established by Farina [1] (for the properties of stable solutions, we refer to the monograph of Dupaigne [19]). Theorem 1.1 (Farina [1]). Let u ∈ C2(RN) be a stable solution of (1.1) with{ 1 < p < +∞ if N ≤ 10 1 < p < pc(N) = (N−2)2−4N+8√N−1 (N−2)(N−10) if N ≥ 11. Then u ≡ 0. On the other hand, forN ≥ 11 and p ≥ pc(N), the equation (1.1) admits a smooth, positive, bounded, stable and radial solution. The exponent pc(N) stands for the Joseph-Lundgren exponent (see [20]). The main tools in [1] are the nonlinear integral estimates combined with the property of stable solutions. In addition, this technique was employed to obtain the optimal Liouville type theorem for finite Morse index classical solutions. Some applications of Liouville type results on qualitative properties of solutions, such as the universal a priori estimate and the behaviour of solution near an isolated singularity, were also studied in [1]. In this paper, we will extend the result of Farina [1] to the general case α > 0 and look for the effect of the degeneracy on the range of the exponent p on Liouville-type 13 Nguyen Thi Quynh theorem. It seems that the presence of the weight term |x|α makes the problem more challenging. The main difficulty is that the Grushin operator is nonautonomous. This requires suitable scaled test functions in the integral estimate. On the other hand, we make use of the properties of the Grushin divergent and the associated distance to derive the nonlinear integral estimates. To our best knowledge, Liouville type theorems for stable solutions to (1.1) have not been established so far. 2. Formulation of main result In this section, we state our main result concerning the nonexistence of nontrivial stable solutions of (1.1). Firstly, without loss of generality, we always assume that the constant a in (1.1) is equal to 1. We then recall the definition of stable solutions, see [19]. Definition 2.1. Let u ∈ C2(RN) be a classical solution of (1.1). The solution u is said to be stable if Qu(ψ) := ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 − p|u|p−1ψ2) dxdy ≥ 0, for all ψ ∈ C1c (RN). (2.1) Recall that Nα := N1 + (α + 1)N2 is the homogeneous dimension of RN1 × RN2 . We define the critical exponent pc(Nα) = { +∞ if Nα ≤ 10 (Nα−2)2−4Nα+8 √ Nα−1 (Nα−2)(Nα−10) if Nα > 10 . (2.2) The main result of this paper is the following: Theorem 2.1. Let u ∈ C2(RN) is a stable solution of (1.1) with 1 < p < pc(Nα). Then, u is the trivial solution. Remark 2.1. • In (2.2), we can see the impact of the exponent α to the critical exponent pc(Nα). • The result of A.Farina-Theorem 1.1 is a consequence of our main result with α = 0. • Note that Theorem 1.1 is optimal in the sense that, for α = 0, N ≥ 11 and p ≥ pc(N), there exist stable radial solutions to (1.1) in RN1 × RN2 (see [1]). In the case α > 0, it seems still difficult to prove the existence of stable solutions to (1.1) on RN1 × RN2 under the condition p ≥ pc(Nα) with Nα > 10. However, if p is sufficiently large, for example p ≥ pc(N1), there exist stable solutions which do not depend on the y-variable. 14 Liouville type theorem for stable solutions to elliptic equations involving the Grushin operator A brief outline of the proof Here we give the outline of the proof inspired by ideas of A. Farina [1]. Suppose that u is a classical stable solution of (1.1). By using the stability condition (2.1) with the test function |u| γ−12 uφ, φ ∈ C∞c (RN), we show that for some γ ≥ 1( p− (γ + 1) 2 4γ )∫ RN |u|p+γφ2dxdy ≤ ∫ RN |u|γ+1|∇Gφ|2dxdy + ( γ + 1 4γ − 1 2 )∫ RN |u|γ+1Gα(φ2)dxdy, (2.3) where ∇G = (∇x, |x|α∇y) is known as Grushin gradient. Let φ = ψm, ψ ∈ C2c (R N ; [−1; 1]) and making use of some integral estimates to arrive at ∫ RN |u|p+γψ2mdxdy ≤ C ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|))p+γp−1 dxdy, (2.4) and∫ RN |u|γ+1Gα(ψ2m)dxdy ≤ C ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|))p+γp−1 dxdy. (2.5) The following key estimate is then deduced from (2.4) and (2.5):∫ RN (∣∣∣∇G (|u| γ−12 .u)∣∣∣2 + |u|p+γ ) ψ2mdxdy ≤ Cp,m,γ ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ| (|∆xψ|+ |x|2α|∆yψ|)) p+γp−1 dxdy. (2.6) Finally, by choosing suitable scaled test functions depending on a large parameter R, the right hand side of (2.6) is bounded by CRNα−2 p+γp−1 . The constant γ is thus taken such that Nα − 2p+γp−1 < 0 and the proof is finished if we let R→ +∞. 3. Proof of Theorem 2.1 We first establish a key tool to prove the main result. The following proposition is an extension of Proposition 4 in [1]. Proposition 3.1. Let p > 1 and u ∈ C2(RN) be a stable solution of (1.1). Fix a real number γ ∈ [ 1, 2p+ 2 √ p(p− 1)− 1 ) and an integerm ≥ p+γ p−1 . Then there is a constant 15 Nguyen Thi Quynh Cp,m,γ > 0 depending only on p,m and γ, such that∫ RN (∣∣∣∇x (|u| γ−12 .u)∣∣∣2 + |x|2α ∣∣∣∇y (|u| γ−12 .u)∣∣∣2 + |u|p+γ ) ψ2mdxdy ≤ Cp,m,γ ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ| (|∆xψ|+ |x|2α|∆yψ|)) p+γp−1 dxdy, (3.1) for all ψ ∈ C2c (RN ; [−1; 1]). Proof. Denote the Grushin gradient by ∇G = (∇x, |x|α∇y) and let φ ∈ C2c (RN). Multiplying (1.1) by |u|γ−1uφ2 and integrating over RN , we obtain∫ RN −Gαu.|u|γ−1uφ2dxdy = ∫ RN |u|p+γφ2dxdy. (3.2) Noticing that ∇x|u|γ+1 = 1γ+1∇xu.u|u|γ−1 and ∇x(|u| γ−1 2 u) = γ+1 2 ∇xu.|u| γ−12 . Using this and the integration by parts to obtain∫ RN (−∆xu).|u|γ−1uφ2dxdy = ∫ RN (∇xu)∇x (|u|γ−1u)φ2dxdy + ∫ RN (∇xu) (|u|γ−1u)∇x(φ2)dxdy = γ ∫ RN |∇xu|2|u|γ−1φ2dxdy + ∫ RN (∇xu) (|u|γ−1u)∇x(φ2)dxdy = γ( γ+1 2 )2 ∫ RN ∣∣∣∇x (|u| γ−12 u)∣∣∣2 φ2dxdy + 1 γ + 1 ∫ RN ∇x|u|γ+1∇x(φ2)dxdy = 4γ (γ + 1)2 ∫ RN ∣∣∣∇x (|u| γ−12 u)∣∣∣2 φ2dxdy − 1 γ + 1 ∫ RN |u|γ+1∆x(φ2)dxdy. The same arguments also follows that∫ RN (−|x|2α∆yu).|u| γ−1 2 uφ2dxdy = 4γ (γ + 1)2 ∫ RN |x|2α ∣∣∣∇y (|u| γ−12 u)∣∣∣2 φ2dxdy − 1 γ + 1 ∫ RN |x|2α|u|γ+1∆y(φ2)dxdy. Inserting the above computations into (3.2), we have Lemma 3.1. There holds∫ RN ∣∣∣∇G (|u| γ−12 u)∣∣∣2 φ2dxdy = (γ + 1)2 4γ ∫ RN |u|p+γφ2dxdy+γ + 1 4γ ∫ RN |u|γ+1Gα(φ2)dxdy. (3.3) 16 Liouville type theorem for stable solutions to elliptic equations involving the Grushin operator We next use the fact that u is a stable solution of (1.1). In (2.1), we choose the test function |u| γ−12 uφ ∈ C1c (RN) and get∫ RN (∣∣∣∇G (|u| γ−12 uφ)∣∣∣2 − p|u|p+γφ2 ) dxdy ≥ 0. (3.4) A straightforward computation gives ∇G ( |u| γ−12 uφ ) = ∇G ( |u| γ−12 u ) φ+ |u| γ−12 u∇Gφ. and∣∣∣∇G (|u| γ−12 uφ)∣∣∣2 = ∣∣∣∇G (|u| γ−12 u)∣∣∣2 φ2 + |u|γ+1|∇Gφ|2 + 1 2 ∇G (|u|γ+1)∇Gφ2. Combining this with the integration by parts, the inequality (3.4) becomes ∫ RN ∣∣∣∇G (|u| γ−12 u)∣∣∣2 φ2dxdy + ∫ RN |u|γ+1|∇Gφ|2dxdy − 1 2 ∫ RN |u|γ+1Gα(φ2)dxdy ≥ ∫ RN p|u|p+γφ2dxdy. (3.5) From Lemma 3.1 and (3.5), we have Lemma 3.2. There holds ( p− (γ + 1) 2 4γ )∫ RN |u|p+γφ2dxdy ≤ ∫ RN |u|γ+1|∇Gφ|2dxdy + ( γ + 1 4γ − 1 2 )∫ RN |u|γ+1Gα(φ2)dxdy. (3.6) Here, the assumptions on p, γ imply that p− (γ+1)2 4γ > 0 and γ+1 4γ − 1 2 < 0. Let m ≥ p+γ p−1 be a fixed integer. For ψ ∈ C2c (RN ; [−1; 1]), put φ = ψm. Hence, |∇xψm|2 = m2|∇xψ|2ψ2m−2; |x|2α |∇yψm|2 = m2|x|2α|∇yψ|2ψ2m−2 and ∆xψ 2m = 2mψ2m−2 ( (2m− 1)|∇xψ|2 + ψ∆xψ ) , |x|2α∆yψ2m = 2mψ2m−2 ( (2m− 1)|x|2α|∇yψ|2 + ψ|x|2α∆yψ ) . 17 Nguyen Thi Quynh The right hand side of (3.6) is smaller than or equal to Cm,γ ∫ RN |u|γ+1ψ2m−2 (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|)) dxdy. Consequently,∫ RN |u|p+γψ2mdxdy ≤ Cm,γ,p ∫ RN |u|γ+1ψ2m−2 (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|)) dxdy. (3.7) Applying Ho¨lder’s inequality to the right hand side of (3.7), we get ∫ RN |u|p+γψ2mdxdy ≤ Cm,γ,p [∫ RN (|u|γ+1ψ2m−2) p+γγ+1 dxdy] 1+γp+γ × [∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|))p+γp−1 dxdy ] p−1 p+γ . (3.8) Moreover, since (2m− 2)p+ γ γ + 1 − 2m = 2m ( p+ γ γ + 1 − 1 ) − 2p+ γ γ + 1 = 2m p− 1 γ + 1 − 2p+ γ γ + 1 ≥ 2p+ γ γ + 1 − 2p+ γ γ + 1 = 0 (3.9) and |ψ| ≤ 1, it implies that (|u|γ+1ψ2m−2) p+γγ+1 = |u|p+γψ(2m−2)p+γγ+1 ≤ |u|p+γψ2m. (3.10) It then follows from (3.8) and (3.10) that ∫ RN |u|p+γψ2mdxdy ≤ Cm,γ,p [∫ RN |u|p+γψ2mdxdy ] 1+γ p+γ × [∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|))p+γp−1 dxdy ] p−1 p+γ , (3.11) or equivalently,∫ RN |u|p+γψ2mdxdy ≤ C p+γ p−1 m,γ,p ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|)) p+γp−1 dxdy. (3.12) 18 Liouville type theorem for stable solutions to elliptic equations involving the Grushin operator Similarly, if we choose φ = ψm, then the second term in the right hand side of (3.3) can be estimated as follows∫ RN |u|γ+1Gα(ψ2m)dxdy ≤ Cm ∫ RN |u|γ+1ψ2m−2 (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|)) dxdy ≤ Cm [∫ RN |u|p+γψ2mdxdy ] 1+γ p+γ × [∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|))p+γp−1 dxdy ] p−1 p+γ ≤ CmC 1+γ p−1 m,γ,p ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|))p+γp−1 dxdy, (3.13) where Cm = 2m(2m− 1) and in the last inequality we have used (3.12). The estimations (3.3) and (3.13) yield ∫ RN ∣∣∣∇G (|u| γ−12 u)∣∣∣2 ψ2mdxdy ≤ C ∫ RN (|∇xψ|2 + |x|2α|∇yψ|2 + |ψ|(|∆xψ|+ |x|2α|∆yψ|)) p+γp−1 dxdy, (3.14) where C depends only on m, p, γ. Finally, Proposition 3.1 results from (3.12) and (3.14). Proof of Theorem 2.1. Let χ ∈ C∞c (R; [0, 1]) have the properties χ(t) = 1 for |t| ≤ 1;χ(t) = 0 for |t| ≥ 2. For R large enough, we choose ψR(x, y) = χ ( |x| R ) χ ( |y| Rα+1 ) ∈ C∞c (RN ; [0, 1]). Then, |∇xψR(x, y)| = 1 R ∣∣∣∣χ′( |x|R )χ( |y|Rα+1 ) ∣∣∣∣ ; |∇yψR(x, y)| = 1R1+α ∣∣∣∣χ( |x|R )χ′( |y|Rα+1 ) ∣∣∣∣ , and |∆xψR(x, y)| = ∣∣∣∣n− 1|x|R χ′( |x|R )χ( |y|Rα+1 ) + 1R2χ′′( |x|R )χ( |y|Rα+1 ) ∣∣∣∣ , |∆yψR(x, y)| = ∣∣∣∣ n− 1|y|R1+αχ( |x|R )χ′( |y|Rα+1 ) + 1R2(1+α)χ( |x|R )χ′′( |y|Rα+1 ) ∣∣∣∣ . 19 Nguyen Thi Quynh These computations together with the boundedness of χ and all its derivatives deduce that (|∇xψR|2 + |x|2α|∇yψR|2 + |ψR| (|∆xψR|+ |x|2α|∆yψR|)) ≤ C1 R2 , (3.15) where C1 is independent of R. Hence, (3.1) and (3.15) give∫ RN (∣∣∣∇x (|u| γ−12 .u)∣∣∣2 + |x|2α ∣∣∣∇y (|u| γ−12 .u)∣∣∣2 + |u|p+γ ) ψ2mR dxdy ≤ ∫ {|x|≤2R;|y|≤2R1+α} (|∇xψR|2 + |x|2α|∇yψR|2 + |ψR| (|∆xψR|+ |x|2α|∆yψR|)) dxdy ≤ C R 2(p+γ p−1 ) RN1RN2(1+α) = CRNα−2 p+γ p−1 , (3.16) where C does not depend on R. Finally, we need to show that, given Nα and p in Theorem 2.1, there is a constant γ ∈ [1, 2p+ 2 √ p(p− 1)− 1) such that Nα − 2p+ γ p− 1 < 0. (3.17) Case Nα ≤ 10: for any p > 1, the function γ 7→ p+γp−1 is continuous, increasing and p+2p+2 √ p(p−1)−1 p−1 > 5p−3 p−1 > 5. Thus, there is γ verifying (3.17). Case Nα > 10: consider the inequality Nα− 2p+ 2p+ 2 √ p(p− 1)− 1 p− 1 < 0 or Nα(p− 1)− 2(p+2p+2 √ p(p− 1)− 1) < 0. (3.18) By changing of variable t = p− 1 > 0, it is equivalent to (Nα − 6)t− 4 < 4 √ t2 + t. (3.19) If t < 4 Nα−6 , then (3.19) is true. 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