VỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾN

VỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾN

Literature review

The partial differential equation was first studied in the mid-18th century in the works of such mathematicians as D’Alembert, Euler, Lagrange and Laplace which is an important tool for describing models of Physics and Mechanics.

Today, after more than two centuries of development, partial differential equations are used in many fields to model many problems in Physics, Mechan- ics, Chemistry, Biology, Economics Due to the complexity of the real problems, the established models are usually nonlinear partial differential equations The study of the existence and qualitative properties of solutions of these classes of equations is one of the main topics of applied mathematical analysis.

One research direction that has attracted the attention of many mathemati- cians in recent years is to consider conditions for the existence or non-existence of solutions through Liouville-type theorems They play a fundamental role and are considered to be the foundation for accessing insights into the structure of the solution set of boundary value problems Liouville-type theorems lead to many particularly important consequences and applications, such as: a priori estimation of the Dirichlet problem, singularity and decay estimates, Liouville- type theorem over half space, estimating universal quantifiers, Harnack-type inequalities, initial burst rates and time decay rates of parabolic problems .

The first topic in this thesis is the study of the porous medium equation/sys- tem with sources u t −∆u m =u p in R N ×R (0.1) and 

In recent years, the Liouville property has emerged as one of the most pow- erful tools in the study of qualitative properties for nonlinear equations As consequences of Liouville type theorems, one can establish a variety of results, for instance, universal, pointwise, a priori estimate; universal and singularity estimates; decay estimates, etc, see[54]and references therein In addition, in [6], the authors have obtained the existence of solutions of semilinear bound- ary value problems in bounded domains by exploiting Liouville type results combined with degree type arguments.

Let us next review some related results in the literature The classification of positive supersolutions of elliptic courterpart of (0.1) and (0.2) has been completely proved, see e.g [3, 17] More precisely, the elliptic counterpart of (0.2) is the Lane-Emden system

−∆v m =u q in R N , (0.3) which has no positive supersolution if and only if

In particular, when p=q, the Lane-Emden system

−∆u m =v p inR N has no positive supersolution if and only ifN−2≤ p/m 2 − 1 On the other hand, the existence and nonexistence of positive solutions to the Lane Emden equation (0.1) has been already established in[34] where the critical exponent is given byp c (N) = N+2 N − 2 However, the same problem for the Lane-Emden system (0.3) has not been completely solved It is known as the Lane-Emden conjecture saying that the system (0.3) has no positive solution if and only if

We refer the readers to the proof of this conjecture in low dimensionsN ≤3 in[50, 57, 58] and in [60] for N =4 The conjecture in the case N ≥ 5 has not been confirmed.

For the parabolic model (0.1) in the semilinear casem=1, the well-known Fujita result ensures the nonexistence of nontrivial nonnegative solutions in

R N ×(0,∞) in the subcritical case 1 < p ≤ N+ N 2 , see [28], [51, Sec 26] In the supercritical case p > N+2 N , problem (0.1) admits, see [37, Example 1], a nonnegative supersolution inR N ×Rof the form u(x,t) 

0 if t≤0, x∈R N , where k,γare suitably chosen.

For the system (0.2) withm=1, Duong and Phan[23]have recently estab- lished optimal Liouville type theorems for nonnegative or positive supersolu- tions Among other things, it is shown in[23]that the system (0.2) withm=1 has no nonnegative supersolution if and only if p,q>0, pq>1 and max §2(p+1) pq−1 ,2(q+1) pq−1 ê

We next consider the problems (0.1) and (0.2) in the quasilinear case m>1. For solutions inR N ×(0,T)of the equations of type (0.1), some local solvability and general regularity results have been studied in [1, 29, 30, 59, 62] It was shown in[30, 56]that whenp≤m+ N 2 , the solutionuof (0.1) inR N ×(0,+∞) with bounded, continuous initial datau 0 ̸≡0 does not exist globally and blow up in a finite time, i.e there is T >0 such that sup x ∈R N u(x,t)→+∞as t →T.

Under the conditionp≤m+ N 2 , it was proved in[16]that any solution of (0.1) inR N ×(0,T)satisfies the following estimate u(x,t)≤C(N,m,p) t − p −1 1 + (T −t) − p 1 −1

This type of estimate has been then proved in [1] for a large range of p, i.e. p

1 are real numbers such that 1 p + 1 q = 1 Then for f ∈L p (R N ) and g∈ L q (R N ), we have

The Grushin operator

We begin this section by recalling some notations Denote by z = (x,y) a generic point ofR N =R N 1 ×R N 2 The usual gradient and the Grushin gradient are respectively denoted by∇ = (∇ x ,∇ y )and ∇ G = (∇ x ,|x| α ∇ y ) The norm ofz associated to the Grushin distance is given by

With this norm, we define the open ball and the sphere of radiusRcentered at z 0 by

We writeB R and∂B R instead ofB(0,R)and ∂B R (0,R).

In what follows, we use the weight function w(z):= |x| 2α

The volume of the ball is computed as, see e.g.,[13, 65],

B R w(z)dz=c(N α ,α)R N α , (1.2) where c(N α ,α)>0 depends onN α andα.

In addition, the co-area formula (see e.g.,[31, Formula 2.4]) implies that

|∇(|z| G )|d H N− 1 , whered H N − 1 is the(N−1)-dimension Hausdorff measure inR N Consequently, the surface area of the sphere is defined by

|∇(|z| G )|d H N− 1 =c(N α ,α)N α R N α −1 These preparations combined with the idea of Garofalo and Lanconelli [31] lead to the definition of the spherical average of a functionV ∈C(R)as follows

|∇(|z| G )|d H N−1 , forR>0 (1.3) Note that whenα=0, this formula becomes the usual spherical average on the Euclidean ball.

The following proposition was known, see[31] However, we give here the proof for the completeness.

Proposition 1.1 Let V ∈C 2 (R N ) Then for every R>0and z 0 ∈R N we have

In particular, when z 0 =0there holds

G α V(z) |z| 2 G − N α −R 2 −N α dz, (1.5) where c(N α ,α)is given in(1.2).

Proof From now on, we denote by Γ(z) =|z| 2 G −N α which is the fundamental solution ofG α , see[31].

It is enough to prove (1.5), (1.4) is then deduced from (1.5) by a simple translation Letϵ >0 be small enough Using integration by parts, we arrive at Z

(1.6) whereνis the outward unit normal vector to∂(B R \ B ϵ ) Sinceν= |∇(|z| ∇|z| G G )| on

G α V dz, (1.7) where in the last equality, we have used the integration by parts On the other hand, a simple computation and (1.1) give

Substituting (1.8) into the second term in (1.6), we get

Inserting (1.7) and (1.9) into (1.6) and usingΓ(z)is the fundamental solution of G α , we obtain

Recalling that|∂B R |=c(N α ,α)N α R N α − 1 , letting ϵ→0 in (1.10), we then have Z

The following proposition shows the relation between the derivative of V and G α V.

Proposition 1.2 Let V ∈C 2 (R N ) Then for every R>0we have

Proof Using the co-area formula, we have

|∇(|z| G )| Γ(z)−R 2−N α d H N− 1 This combined with (1.5) imply that

|∇(|z| G )|R 1 −N α d H N − 1 , where in the last equality, we have usedΓ(z) = |z| 2−N G α =R 2 −N α on ∂B R This equality and the co-area formula imply (1.12).

The following result is the Hardy type inequality involving the Grushin op- erator, see[41].

Lemma 1.1 " Let r,s∈Rbe such that N α +2> r−s and N 1 >2α−s.Then for everyϕ∈C c 1 (R N ), we have

The fractional Laplacian

Assuming 0 m > 1 The problem (2.1) has no nontrivial nonnegative weak supersolution inR N ×Rif and only if p≤ mN+ N 2

In the case p > mN N +2 , we shall construct explicitly nontrivial nonnegative weak solutions of (2.1) (see the last part of the proof of Theorem 2.1).

The second purpose in this chapter is to establish the nonexistence result for the system (2.2).

Theorem 2.2 Assume that p,q > m> 1 The problem(2.2) has no nontrivial nonnegative weak supersolution inR N ×Rif and only if

When the condition (2.3) fails, we shall also construct nontrivial nonnega- tive weak solutions to (2.2).

Remark 2.1 Notice that when m=1, (2.1) and (2.2) becomes u t −∆u=u p inR N ×R and 

The critical exponents in Theorem 2.1 and Theorem 2.2 are respectively given by

These exponents were found in[23] In this case, our results coincide with that in[23].

Remark 2.2 From our result, we want to address a question on the nonex- istence of nontrivial nonnegative solutions of (2.1) and (2.2) when p < m or q 1 complicates the method and requires careful parameter selection in the scaling argument Our results are sharp due to the constructions of nontrivial nonnegative weak supersolutions of (2.1) and (2.2).

2.2.1 Nonexistence result for the porous medium equation

Let us begin this section by proving the nonexistence result In what follows, we denote byC a generic constant which may change from line to another and independent of solutions Set β = N(m−1) +2p p−m+1 For r>0, define

Letψ∈C c ∞ (R N ×R;[0, 1])be a test function satisfyingψ=1 onB 1 andψ=0 outsideB 2 Forr >0, putψ r =ψ l ( x r , r t β )where l is chosen later on.

Suppose thatuis a nonnegative weak supersolution of (2.1) By the defini- tion of supersolutions above withφ =ψ r , we have

|∂ t ψ r | ≤ C r β ψ r l −1 l This observation combined with (2.4) and the fact that 0≤ψ r ≤1 leads to Z

Next, applying the Hửlder inequality to the first term in the right hand side of (2.5), we arrive at

Similarly, the second term in the right hand side of (2.5) is controlled as follows Z

Let us choosel large enough such that (l− ml 2 )p >1 and (l − l 1 )p >1 Thus, substi- tuting (2.6) and (2.7) into (2.4), we obtain

Remark that, by the definition of β above and some elementary computation, we have

Thus, we deduce from (2.8) that

We next consider two cases ofκ.

It is easy to see that this condition is equivalent to p< mN+2

By simplifying the inequality (2.9) and then lettingr →+∞, one has

This implies thatuis the trivial nonnegative supersolution.

As above,κ=0 is equivalent to p= mN+2

R N ×R u p d x d t is finite However, this consequently yields the fact that the right hand side of (2.9) tends to 0 as r →+∞ There- fore, we again obtain from (2.9) that

This is the case if only ifu=0.

The rest of the proof is devoted to the existence result For p > mN+ N 2 , we construct a nontrivial nonnegative weak supersolution of the form u(x,t) 

, hereϵ is a small positive constant, γ= 2 p−m ( p − 1 ) and t + = max(t, 0) Indeed, on the range t >0 andϵ− γ(m 2m − 1) |x| t 2 2γ + 1 >0, we compute

= (γN− 1 p−1)ϵ − m−1 p −1 u p (x,t). Since p> mN N + 2 , there holdsγN − p− 1 1 >0 Let us choose ϵsmall enough such that(γN− p − 1 1 )ϵ − m−1 p−1 >1 Then, we have shown thatu(x,t)constructed above is a nontrivial nonnegative weak supersolution of (2.1).

2.2.2 Nonexistence result for the porous medium system

In this section, we divide the proof of Theorem 2.2 into two parts: nonexis- tence result and existence of solutions.

The nonexistence proof relies on the rescaled test-function method However, when m > 1 (quasilinear case), the method becomes intricate and demands precise computations For such cases, the existence of a nonnegative weak supersolution (u, v) to (2.2) and a test function ψ ∈ are necessary.

C c ∞ (R;[0, 1])be a test function satisfyingψ=1 on[−1, 1]and ψ=0 outside [−2, 2] For r>0, denote byψ r =ψ l ( r x β )ψ l ( r t α )wherel is chosen later on and α,β >0 In what follows, we use the notation

Suppose that (u,v)is a nonnegative weak supersolution of (2.2) Then we have

By using similar arguments as in (2.8), we also obtain from (2.10) that

It results from the definition of supersolutions above with the test functionψ k r , k= ( l − ml 2 ) q that

B 2r u q ψ k r d x d t (2.12) Again, similar to (2.8), we also arrive at

(2.13) where in the last inequality we have chosen l large such that (kl − kl 1)p > 1 and

(l k− 2 )q mkl >1 Combining (2.11) and (2.13), we deduce that

For simplicity of notation, we put κ 1=−αq+ (q−1)(βN+α)−α+(p−1)(βN +α) p , κ 2=−αq+ (q−1)(βN+α)−2β+(p−m)(βN +α) p , κ 3=−2βq m +(q−m)(βN +α) m −α+ (p−1)(βN +α) p andκ 4=−2βq m +(q−m)(βN +α) m −2β+ (p−m)(βN +α) p

Then, as a consequence of (2.14), there holds

Suppose that the four powers ofr in the right hand side of (2.15) are nonposi- tive, i.e.,κ i ≤0 fori=1, 2, 3, 4 Then, arguing as in the proof of Theorem 2.1, we obtain from (2.15) that

The rest of the proof of nonexistence result is devoted to showing that, with suitable α,β > 0, the four powers of r in the right hand side of (2.15) are nonpositive We first have some elementary properties of these powers.

Without loss of generality, we may assume that p≥q Then, max

The following lemma gives the condition ensuring the nonpositivity ofκ i , i 1, 2, 3, 4.

Lemma 2.1 Under the condition p ≥ q > m > 1 and α,β > 0, the following assertions hold:

(i) κ 1≤0if and only if N ≤ α β pq− p+1 1

(ii) κ 2≤0if and only if N ≤ 2p+ β α m pq−m (iii) κ 3≤0if and only if N ≤ 2pq− α β (pq−m(p+ 1 )) pq−m

(iv) κ 4≤0if and only if N ≤ 2 (pq+pm)− α β (pq−m 2 ) pq−m 2

Proof The lemma follows from some careful computation We omit the details here.

The next lemma gives some choices of αand β such that the conditions of

N in Lemma 2.1 attain the sharp conditionN ≤ p(q+1)− 2 (p+ m(p+1) 1 )

Lemma 2.2 Under the hypothesis of Lemma 2.1, we have

(iii) 2pq− α β (pq−m(p+1)) pq−m = p(q+1)−m(p+1) 2 ( p + 1 ) iff α β = p(q+1)−m(p+1) 2 ( pq − 1 )

(iv) 2(pq+pm)− α β (pq − m 2 ) pq−m 2 = p(q+1)−m(p+1) 2 ( p + 1 ) iff α β = 2 ( pq−m pq + pm 2 ) − p(q+1)−m(p+1) 2 ( pq − 1 )

2(pq+pm) pq−m 2 − 2(pq−1) p(q+1)−m(p+1) ≤ 2(pq−1) p(q+1)−m(p+1)

Proof The proof of this lemma is done thanks to some elementary computation.

We also omit the details.

Finally, combining Lemmas 2.1, 2.2 and (2.3), we choose α β = 2(pq−1) p(q+1)−m(p+1) to deduce that κ i ≤ 0 for i = 1, 2, 3, 4 The proof of nonexistence result is complete.

We shall construct a nontrivial nonnegative weak supersolution of (2.2) as fol- lows First, we put u(x,t) 

, whereϵ,α 1,α 2,γ 1andγ 2are positive parameters chosen later on By using the same computations as in the last part of the proof of Theorem 2.1 and choosing α 1+1=mα 1+2γ 1=qα 2, (2.17) we obtain on the domain{(x,t);u(x,t)>0}that u t −∆u m ≥(γ 1 N −α 1)t −α 1 − 1 ϵ−γ 1(m−1)

Similarly, by taking α 2+1=mα 2+2γ 2=pα 1, (2.19) we also arrive at, after some straightforward computation, v t −∆v m

(2.20) on the set{(x,t);v(x,t)>0} Thanks to (2.17) and (2.19), we deduce that α 1= q+1 pq−1, α 2= p+1 pq−1,

It is easy to see thatγ 1≥γ 2provided p≥q>m>1 Moreover, it follows from the condition (2.16) thatγ 2 N−α 2 >0 andγ 1 N −α 1 >0.

Notice, on the other hand, that by choosing ϵ < γ 1 (m 2m − 1) , we have u(x,t) v(x,t) =0 when t≤1 Letχ be the characteristic function of the set

Set U(x,t) = u(x,t)χ(x,t) and V(x,t) = v(x,t)χ(x,t) Hence, on the do- mainΩ, we deduce from (2.18) and (2.20) that

It is sufficient to chooseϵ small enough such that

Thus, we obtain a nontrivial nonnegative weak supersolution (U,V) of the system whenϵsmall enough The proof is complete.

In this chapter, we proved the nonexistence of nontrivial nonnegative weak supersolutions for the porous medium equations/systems with sources in the subcritical case by using the test function method Nevertheless, due to the presence ofm>1, this method becomes more complicated and it also requires a suitable choice of parameters in scaling argument.

In the super critical case, we shall also construct explicitly nontrivial non- negative weak solutions to the equation/system This construction is based on the heat kernel of the porous medium equation In the particular case, when m=1 our result recovers some results in[23].

On stable solutions of a weighted degenerate elliptic equation

Problem setting and main result

In this chapter, we consider the following equation

−G α u+c(z)ã ∇ α u=h(z)e u ,z= (x,y)∈R N 1 ìR N 2 =R N , (3.1) where G α =∆ x + (1+α) 2 |x| 2α ∆ y , α >0, is the Grushin operator, the weight functionh(z)is continuous Here,c(z)is a vector field satisfying div G c=0 andβ :=sup

The Laplace operator can be seen as a special case of the Grushin operator with α = 0 When α > 0, G α is elliptic for |x| ̸= 0 and degenerates on the manifold{0} ×R N 2 This operator was introduced in [4, 35]and has attracted the attention of many mathematicians It is well-known that the operator G α belongs to a wide class of subelliptic operators studied by Franchi et al in [27] In the special case α= 1, N 1 =2n, N 2 =1, the corresponding operator appeared in the study of the Cauchy-Riemann Yamabe problem with constant Webster scalar curvature in the Heisenberg groupC n ×R.

It is worthy mentioning that a class of elliptic systems with gradient term ap- pears when considering electrochemical models in engineering and some other models in fluid dynamics We refer to[15]and [8]for more details.

In[10], the author obtained some classification of stable positive solutions to the equation

−∆u+cã ∇u=u p inR N , where c is a smooth, divergence free vector field satisfying |c(x)| ≤ 1 +|x| ϵ , ϵ is small enough The technique used in [10] is a combination of test function method and the generalized Hardy inequality[9].

In the case of exponential nonlinearity, by exploiting the technique in [10], the authors in[38] established the nonexistence of stable solutions to

−∆u+cã ∇u=e u inR N when 3≤N ≤9 and c satisfies the same conditions as in[10].

The purpose of this chapter is to generalize some results in [19, 38] to the weighted degenerate elliptic equation (3.1) Let us first recall the definition of stability see e.g [10, 38].

−G α u+cã ∇ α u=h(z)e u is called stable if there exists a positive function F ∈C 2 (R N )such that

−G α F +cã ∇ α F ≥h(z)e u F (3.3)Recall that N α := N 1 + (1+α)N 2 is the homogeneous dimension ofR N as- sociated to the Grushin operator.

The main result of this chapter is the following.

Theorem 3.1 Suppose that the weight function h is continuous and h(z)≥C|z| l G for some l≥0 If

, then(3.1)has no stable solution.

Remark 3.1 Some remarks are in order.

• Notice that when l = 0, α = 0 and β small enough, then we recover the nonexistence result in[38] from Theorem 3.1 where the condition is merely 3≤N ≤9.

(N −2)(10−N), Theorem 3.1 implies the nonexistence result in[19].

• Inspired by the question in [10], one conjectures that the condition onβ can be removed Nevertheless, this is an open question, even in the case of Laplace operator.

Let us now introduce the idea of the proof The proof is based on the energy method, the test function method together with the Hardy inequality Neverthe- less, we need to construct a suitable test function corresponding to the Grushin operator Moreover, to remove the smallness condition in[38], we need to use some delicate estimates to obtain a relax condition onβ.

Proof of nonexistence of stable solutions

In this section, we prove our result by contradiction In what follows, we denote by C a generic constant which may change from line to another By multiplying (3.1) by test functionϕ∈ C c ∞ (R N )and then integrating overR N , one gets

Using integration by parts, we have

F and then integrating overR N , we get Z

The first term in the left hand side is

Using the Hardy inequality, we obtain

|∇ α ψ| 2 e 2tu +2∇ α e tu ãe tu ∇ α ψ+|∇ α e tu | 2 ψ 2 dz

Letχ be a test function inC c ∞ (R)such that χ(t) 

ForRlarge enough, we consider the test function ψ R =χ

Then, it is easy to see that∇ α ψ R =0 outside the set

R 2 ψ 2m R − 2 , where m>2 is chosen later Then, (3.9) withψreplaced byψ m R becomes

Using the Hửlder inequality, we get

2t 1 +1 , where mis chosen such that(m−1)(2t+1)/t >2m Then (3.10) implies

Thus (3.11) and the above computation give

We next show that there exists t>0 such that

2l+4 Consequently, there exists t >0 such that

2(N α −2) 2 β 2 + (N α −2) 2 >t and t> N α −2 2l+4. Therefore we obtain (3.14) and (3.15) Finally, lettingR→ ∞in (3.12), we get a contradiction The proof is complete.

In this chapter, we studied the elliptic equation involving the Grushin oper- ator, advection term and exponential nonlinearity By using the test function method combined with stability inequality and nonlinear integral estimates, we obtained the nonexistence of stable solutions of the equation.

Our result is an extension of some results in[19, 38]to the case of weighted degenerate elliptic equation.

On the nonexistence result for the nonlinear fractional Choquard

Problem setting and main results

In this chapter, we are interested in the classification of stable solutions of the fractional Choquard equation on the whole spaceR N

|x| N−2s ∗u p u p−1 , (4.1) where 0 < s < 1 and p ∈ R Here “∗” stands for the convolution of two functions, i.e.

In the case s = 1, i.e the fractional Laplacian is replaced by the Laplace operator, the problem (4.1) becomes

|x| N−2 ∗u p u p−1 which belongs to the class of static Choquard equation

The Choquard equation arises in the Hartree-Fock theory of the nonlinear Schrửdinger equations, see e.g.,[45–47] Recently, the equations of Choquard type have been attracted much attention, see e.g., [33, 42, 44, 63] and refer- ences given there We refer the readers to[5, 48, 49]for the recent results on the existence, symmetry and regularity of solutions of the Choquard type equa- tions In particular, an introduction of mathematical treatment to the Choquard equations is provided in the survey article[53].

A natural question arises from[39] that whether the problem (4.1) admits a positive solution when p ≤1 or not Recall that the answer to this question in the local case s = 1 was given in [42] In this chapter, we also give the answer in the nonlocal case 0 < s < 1 Here, we consider positive solutions u∈C 2 σ (R N )∩ L s (R N ), for someσ >s, in the classical sense.

We next address a question on the nonexistence of positive stable solutions of (4.1) involving the fractional Laplacian For this purpose, we consider the class of positive solutionsu∈C 2σ (R N ), for someσ >s, satisfying

(1+|x|) N+ 2s d x