Liouville theorems of solutions to mixed order Hénon-Hardy type system with exponential nonlinearity

1.1 Background and main results

This paper is devoted to the study of Liouville type results for nonnegative solutions to the mixed order Hénon-Hardy type system with exponential nonlinearity and Dirichlet boundary conditions on a half space R + 2 :

where 0 < α < 2, R + 2 = x = ( x 1 , x 2 ) ∈ R 2 | x 2 > 0 is the upper half Euclidean space. Throughout this paper, we always assume that p ₁, p ₂, q ₁, q ₂ > 0, a > −α and b > −2.

Recall that the nonlocal fractional Laplacians ( − Δ ) α 2 (where 0 < α < 2) in R n is defined by

for any functions u ∈ C loc 1,1 ∩ L α ( R n ) , where the constant C α , n = ∫ R n 1 − cos ( 2 π ζ 1 ) | ζ | n + α d ζ − 1 , P.V. stands for the Cauchy principle value and the function spaces

In this space, the fractional Laplacians can also be defined by the Fourier transform

On the other hand, the fractional Laplacians ( − Δ ) α 2 can also be defined equivalently (see [1]) by Caffarelli and Silvestre’s extension method (see [2]) for u ∈ C loc 1,1 ( R n ) ∩ L α ( R n ) .

Throughout this paper, we assume the classical solution

The fractional Laplacian has attracted much attention due to its practical applications in many fields. For example, it has been frequently used to model diverse physical phenomena, such as turbulence, water waves, anomalous diffusion, phase transitions, flame propagation, quasi-geostrophic flows and so on (see [3]–[6] and the references therein). It also has several applications in probability, optimization and finance (please see [7]).

Mathematically, there have been extensive interests in conformal geometry problems and Sobolev inequalities. A well-known example is the following Dirichlet problem:

PDEs of the form (1.4) are called the fractional order or higher order Hénon, Lane-Emden, Hardy equations for a > 0, a = 0, a < 0, respectively. The equation of (1.4) arises from the conformal problems and has also attracted much attention in the literature. For instance, Fall and Weth [8] established the nonexistence of positive solutions u ∈ to problem (1.4) via extension methods. After then, Quaas and Xia [9] showed that there are no positive viscosity bounded solutions of (1.4) by using the method of moving planes. In the case of a = 0, Chen, Fang and Yang [10] established Liouville theorem for Dirichlet problem (1.4) when 1 < p ≤ n + α n − α by using the method of moving planes in integral forms. Lu and Zhu [11] derive axial symmetry and regularity of solutions to integral equations with general nonlinearity in half space. Later on, Chen, Li and Zhang [12] reproved the Liouville theorem in [10] via a quite different and much simpler approach-a combination of direct methods of moving spheres and moving planes. Dai and Qin [13] first presented the basic idea of method of scaling spheres, as a preliminary application, they derived Liouville type results for problem (1.4) when 1 ≤ p ≤ n + α + 2 a n − α and a > −α. Subsequently, Dai and Qin [14] applied the method of scaling spheres to establish the Liouville theorem for critical order Hénon-Lane-Emden equation (1.4) (where α = n) with the Navier boundary condition in the cases −n < a < +∞ and 1 ≤ p < +∞. One should notice that, the subcritical condition (on singularities) on nonlinearity f(x, u) introduced in [14] is entirely different from the subcritical condition (on the growth of u) in previous literature, which indicates that the method of scaling spheres is quite suitable for dealing with problems with singularities. The method of scaling spheres takes full advantage of the integral representation formula, and yields the Liouville type results from the scaling spheres procedure (w.r.t. the singularities given by the problems), a bootstrap technique and the a priori lower bound estimates on asymptotic behavior (w.r.t. the singularities). Consequently, it can be applied conveniently to various problems (PDEs, IEs, Systems) without translation invariance or with singularities. For more applications of the method of scaling spheres on R n , R + n , B _R and R n \ B R , c.f. [15]–[19] and the references therein. For other related results on Liouville type theorems, we refer to [20]–[29] and the references therein.

One should note that, even for equations involving classical second order Laplace operators, there are barely a few Liouville type results on general domains (such as cone-like domains) except on R n , R + n and bounded star-shaped domains with regular boundary (such that the formula of integration by parts holds). Very recently, in [30], Dai and Qin introduced and applied further the method of scaling spheres as a unified approach to optimal Liouville theorems for fractional, second and higher order problems (without translation invariance or with singularities) on general bounded or unbounded MSS applicable domains including any cone-like domains, while Liouville theorems in previous literatures are mainly focused on the whole space or half space. A domain is MSS applicable if and only if either itself or its complement is radially convex. Besides leading to the boundedness of Palais-Smale sequences and hence existence of solutions on MSS applicable domains via variational methods, these Liouville theorems on general domains, in conjunction with the blowing-up arguments, can yield a priori estimates and hence existence of solutions to non-variational boundary value problems on bounded domains or on Riemannian manifolds with non-C ¹ smooth boundaries. Recall that, in previous works, the C ¹-smoothness of boundary was required to guarantee that the limiting shape of the scaling domain is the whole space or half space, so that only Liouville theorems in whole space or half space are involved to get a contradiction.

Another problem we are interested in is the Hénon-Hardy type equations with exponential nonlinearity, namely

The problem with exponential nonlinearity like (1.5) finds their origin in problems of physics and geometry. For example, it can be used as a model for the expansion of Universe with isothermal gas sphere in gravitational equilibrium (see [31]). Eq. (1.5) can describes the space charge of electricity around a glowing wire (see [32]). From the viewpoint of mathematics, there is a lot of results have also been achieved on problem (1.5). For instance, Dancer and Farina [33] proved that problem (1.5) with a = 0 admits classical stable at infinity solutions if and only if n = 2 or n ≥ 10. Subsequently, Wang and Ye [34] derived (1.5) does not admit any stable solution provided that a > −2 and 2 ≤ n < 10 + 4a. Later on, Guo et al. [35] proved that problem (1.5) does not admit any stable at infinity solution provided that 10 ≤ n < 10 + 4a and a > 0. Recently, Fazly et al. [36] derived the non-existence of finite Morse index solution to the following nonlocal Hénon-Gelfand-Liouville problem:

where s ∈ (0, 1) and a > 0. For more kinds of literature on the quantitative and qualitative properties of solutions to fractional order or higher order problems, please refer to [15]–[17], [20], [37]–[39], [13], [14], [18], [19], [23], [40]–[42], [25], [43], [27], [44]–[46] and the references therein.

A natural question is one may ask what will happen if we combine the above equation (1.4) and (1.5) in half space together? In this paper, our main purpose is to study the properties of the solutions of mixed order Hénon-Hardy type system with exponential nonlinearity (1.1).

Generally speaking, dealing with nonlocal problems is more challenging than local problems due to the nonlocal nature of the fractional operator. There are several methods for dealing with non-local problems as follows. Caffarelli and Silvestre [2] define the nonlocal operators via the extension method and reduce the nonlocal problem into a local problem in higher dimensions. Later on, Chen and Li (see [1], [47]) give another approach, which considers the equivalent IEs instead of PDEs by deriving the integral representation formulas of solutions to the equations involving fractional operators, we refer to [18], [19], [21], [48]–[50] and references therein for a series of fruitful results in elliptic equations and systems. Subsequently, Chen et al. developed a direct method of moving planes [51] and direct sliding method [22]. These two direct methods are also applied to other non-local operators, such as fractional p-Laplacians ( − Δ ) p s in [22], [52], [53], pseudo-relativistic Schrödinger operators ( − Δ + m 2 ) s (see [54]–[56]), uniformly elliptic Bellman integro-differential operator F _s (see [57], [58], c.f. arXiv:2004.02879v3 for corrections to [57]), tempered fractional operator − Δ + λ α 2 (see [59]), and so on.

In this paper, we first need to show that the equivalence between PDEs system (1.1) and the following integral system (IEs system):

where the Green’s function G α + ( x , y ) associated with Dirichlet problem for ( − Δ ) α 2 (α ∈ (0, 2)) on R + 2 is given by

and G ⁺(x, y) denotes the Green’s function for (−Δ) on R + 2 , which is

where x ̄ ≔ ( x 1 , − x 2 ) is the reflection of x with respect to the boundary ∂ R + 2 .

Our equivalence results are as follows.

Next, we can consider the IEs system (1.6) instead of PDEs system (1.1). By applying the method of scaling spheres introduced in [13], we derive the following Liouville result of the solution (u, v) to the equivalent IEs system (1.6). Our main result for the equivalent IEs system (1.6) is the following theorem.

Combining Theorem 1.3 with the equivalence between the PDEs system (1.1) and the IEs system (1.6), we obtain immediately the Liouville result of the classical solutions (u, v) to the mixed order Hénon-Hardy type system with exponential nonlinearity (1.1). Our main result for PDEs system (1.1) is the following theorem.

1.2 Extensions to general nonlinearities

Consider the following integral system for general mixed order elliptic equations on R + 2 :

where ( u , v ) ∈ L α ( R 2 ) ∩ C loc 1,1 R + 2 ∩ C R + 2 ̄ ∩ C 2 R + 2 ∩ C R + 2 ̄ and the nonlinear terms f , g : R + 2 × R + ̄ × R → R + ̄ .

We also suppose that the nonlinearity f and g satisfies the following assumptions:

1. f(x, s, t), g(x, s, t) are nondecreasing in (s, t);

2. There exist σ < α and θ < 2 such that, |x| ^σ f(x, u, v) are locally Lipschitz on u and v in R + 2 × R + ̄ × R ̄ , |x| ^θ g(x, u, v) are locally Lipschitz on u and v in R + 2 × R + ̄ × R ̄ ;

3. There exist a cone C = x ∈ R + 2 | x 2 > | x | 2 ⊂ R + 2 containing the positive x ₂-axis with vertex at 0, constants C > 0, −α < a < +∞, −α < b < +∞, 0 < p 1 ≤ 2 a + α + 2 2 − α , 0 < p 2 ≤ 4 + 2 b 2 − α , 0 < q ₁ < +∞ and 0 ≤ q ₂ < +∞ such that, the nonlinear term

Using the same technique (the method of scaling spheres introduced in [13]) to the generalized IEs system (1.11), we can derive the following Liouville type result for the IEs system (1.11).

In fact, IEs system (1.11) are closely related to the following PDEs system:

The equivalence between the generalized PDEs system (1.14) and IEs system (1.11) can be proved in entirely similar way as the proof of Theorem 1.1 under the assumptions p 1 ≥ − 2 a α − 1 and q ₂ > 0. We omit the details here.

As a consequence of Theorem 1.7 and the equivalence, we derive the following Liouville-type result for the generalized PDEs system (1.14).

The rest of our paper is organized as follows. In Section 2, we will carry out our proof of Theorem 1.1. After we have this important property of the solutions, we will present the proof of Theorem 1.3 via the method of moving spheres in Section 3.

In what follows, we will use the same C to denote a constant of which value may be different from line to line, and only the relevant dependence is specified.