Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications
Introduction
In this paper, we first establish Liouville theorem for the following n-th order Hénon-Lane-Emden equations with Navier boundary conditions: where is the upper half Euclidean space, , is even, and .
For , PDEs of the form are called the fractional order or higher order Hénon equations for , Lane-Emden equations for , and Hardy equations for . The nonlinear terms in (1.2) are called critical if ( if ) and subcritical if . These equations have numerous important applications in conformal geometry and Sobolev inequalities. In particular, in the case , (1.2) becomes the well-known Lane-Emden equation, which models many phenomena in mathematical physics and astrophysics. When , equations of type (1.2) was first proposed by Hénon in [29] when he studied rotating stellar structures.
We say that equations (1.2) are in critical order if and are in subcritical order if . Being essentially different from the subcritical order operators, in the critical order case, the fundamental solution of is , which changes signs. Consequently, one can't deal with critical order case by simply employing similar arguments as subcritical cases. For instance, in critical order case, the integral representation in terms of the fundamental solution cannot be deduced directly from the super poly-harmonic properties of the solutions.
Liouville type theorems (i.e., nonexistence of nontrivial nonnegative solutions) and related properties for equations (1.2) in the whole space and the half space have been extensively studied (see [1], [4], [6], [7], [9], [14], [16], [19], [20], [21], [22], [23], [24], [25], [27], [31], [32], [33], [36], [37], [41], [42], [43], [44], [45], [46] and the references therein). These Liouville theorems, in conjunction with the blowing-up and re-scaling arguments, are crucial in establishing a priori estimates and hence existence of positive solutions to non-variational boundary value problems for a class of elliptic equations on bounded domains or on Riemannian manifolds with boundaries (see [4], [11], [21], [28], [40]).
(i) Whole space
For Liouville type results of equations (1.2) in in the subcritical order cases , we refer to [1], [9], [14], [16], [19], [20], [21], [22], [23], [25], [27], [31], [32], [36], [37], [41], [46] and the references therein. In the critical order case , Liouville theorems for equations (1.2) have been established in [44] for and in [4] for in the whole space . For the special case , their results can be concluded as the following theorem. Theorem 1.1 ([4], [44]) Suppose is an even integer, and u is a nonnegative classical solution of Then, we have in .
(ii) Half space
(a) Subcritical order cases . The Liouville type results concerning equations (1.2) on in subcritical order cases are too numerous, here we only list some of them.
For , there are many works on the Liouville type theorems for Lane-Emden equations on half space , for instance, see [6], [7], [14], [15], [21], [24], [34], [42], [43] and the references therein. Reichel and Weth [42] proved Liouville theorem in the class of bounded nonnegative solutions for Dirichlet problem of higher order Lane-Emden equations (1.2) (i.e., and with ) on in the cases , subsequently they also derived in [43] Liouville theorem for general nonnegative solutions in the cases . In [6], Chen, Fang and Li established Liouville theorem for Navier problem of Lane-Emden equation (1.2) on in the higher order cases with and . In a recent work [21], Dai and Qin developed the method of scaling spheres, which becomes a powerful tool in deriving asymptotic estimates of solutions and Liouville type theorems for problems without translation invariance. As one of many immediate applications, they established in [21] the Liouville theorem for subcritical order Hénon-Hardy type IEs and Lane-Emden type PDEs with Navier boundary conditions on for all .
For Liouville theorems on , the cases have not been fully understood. For , by using the method of scaling spheres developed initially in [21], Dai, Qin and Zhang [24] proved the Liouville theorem for subcritical order Hénon-Lane-Emden equations (1.2) (i.e., with ) with Navier boundary conditions on in the cases .
(b) Critical order case . To the best of our knowledge, so far there have not seen any results.
In this paper, by applying the method of scaling spheres in integral forms, we will establish Liouville theorem for the Navier problem of critical order Hénon-Lane-Emden equation (1.1) on in all the cases that , and . Our theorems seem to be the first results on Navier problems for critical order equations on .
It's well known that the super poly-harmonic properties of solutions are crucial in establishing Liouville type theorems and the integral representation formulae for higher order or fractional order PDEs (see e.g. [4], [5], [6], [19], [20], [24], [31], [46]). In order to prove the equivalence between PDE (1.1) and corresponding integral equation, we will first prove the following generalized theorem on super poly-harmonic properties, namely, we allow and assume that if . Theorem 1.2 (Super poly-harmonic properties) Assume is even, , and u is a nonnegative solution of (1.1). If one of the following two assumptions holds, then for every and all .
Based on the above super poly-harmonic properties, we can deduce the equivalence between PDE (1.1) and the following integral equation where denotes the Green function for on with Navier boundary conditions, and is the reflection of x with respect to the boundary . That is, we have the following theorem. Theorem 1.3 Assume is even, and . If u is a nonnegative classical solution of (1.1) satisfying one of the two assumptions in (1.3), then u is also a nonnegative solution of integral equation (1.4). If u is a nonnegative classical solution to integral equation (1.4), then u also solves (1.1).
Next, we consider the integral equations (1.4) instead of PDE (1.1). We will study the integral equation (1.4) via the method of scaling spheres in integral forms developed by Dai and Qin in [21].
The method of scaling spheres is essentially a frozen variant of the method of moving spheres, that is, we only dilate or shrink the spheres with respect to one fixed center (say, the singular point). The method of moving spheres was initially used by Padilla [39], Chen and Li [10]. It was developed subsequently by Li and Zhu [34], in which calculus lemmas and classification results were proved. In order to derive classification results or Liouville type theorems, the method of moving spheres usually need to move spheres centered at every point in or synchronously (i.e., either all stop at finite radius or all stop at +∞) in conjunction with calculus lemmas and ODE analysis. Later, it was further developed by Li [32], Chen and Li [12], Li and Zhang [35], Jin, Li and Xu [30] and Chen, Li and Zhang [14]. One should note that, being different from the method of moving spheres, the method of scaling spheres takes full advantage of the integral representation formulae of solutions and can be applied to various problems with singularities or without translation invariance on general domains. It can also be applied to various fractional or higher order problems in the cases that the method of moving planes in conjunction with Kelvin transforms do not work (see [21], [24]).
The method of scaling spheres used in this paper consists of two main steps. First, by combining the properties of Green's function (see Lemma 4.1), the integral representation formulae and the scaling spheres procedure, we will show that positive solutions can be bounded from below by a positive constant for large in a cone . Second, based on the first step, a “Bootstrap” iteration process in conjunction with the integral representation formulae will provide better and better lower bound estimates on the asymptotic behavior of the positive solution u in (see Theorem 4.2), which finally lead to a contradiction with the integrability of solutions indicated by the integral representation formulae unless the solution (see Section 4).
Our Liouville type result for IE (1.4) is the following theorem. Theorem 1.4 Assume , and . If is a nonnegative solution to (1.4), then .
Remark 1.5 Note that we do not assume n to be even in Theorem 1.4. It is also unexpected that the above Theorem 1.4 still holds for . One can see clearly from the proof that the assumption in Theorem 1.4 can be weakened to for some small .
As a consequence of Theorem 1.3, Theorem 1.4, we obtain immediately the following Liouville type theorem on PDE (1.1). Theorem 1.6 Assume is even, and . Suppose is a nonnegative classical solution to (1.1), then .
As an immediate application of the Liouville theorems (Theorem 1.1 for and Theorem 1.6 for ), we can derive a priori estimates and existence of positive solutions to critical order Lane-Emden equations in bounded domains Ω for all .
In general, let the critical order uniformly elliptic operator L be defined by where is even and the coefficients and such that there exists constant with Consider the Navier boundary value problem: where is even, and Ω is a bounded domain with boundary .
By virtue of the Liouville theorem in established in [4], [44] (see also Theorem 1.1) and Liouville theorem in (Theorem 1.6), using entirely similar blowing-up and re-scaling methods as in the proof of Theorem 6 in [6] by Chen, Fang and Li, we can derive the following a priori estimate for classical solutions (possibly sign-changing solutions) to the critical order Navier problem (1.8) in the full range . Theorem 1.7 Assume is even, and there exist positive, continuous functions and : such that uniformly with respect to . Then there exists a constant depending only on Ω, n, p, , , such that for every classical solution u of Navier problem (1.8).
Remark 1.8 The proof of Theorem 1.7 is entirely similar to that of Theorem 6 in [6] (see also Theorem 1.13 in [21]). We only need to replace the Liouville theorems for subcritical order Lane-Emden equations in (see Lin [31] for fourth order and Wei and Xu [46] for general even order) by Liouville theorems for critical order equations in (see Souto [44] for and Chen, Dai and Qin [4] for , see also Theorem 1.1), and replace the Liouville theorems for subcritical order Lane-Emden equations on (Theorem 5 in [6], or further, Theorem 1.20 in [21]) by Theorem 1.6 in the proof. Thus we omit the details of the proof.
One can immediately apply Theorem 1.7 to the following critical order Navier problem: where , is a bounded domain with boundary and t is an arbitrary nonnegative real number.
We can deduce the following corollary from Theorem 1.7. Corollary 1.9 Assume . Then, for any nonnegative solution to the critical order Navier problem (1.11), we have
Remark 1.10 In [4], due to the lack of Liouville theorem in (Theorem 1.6), the authors first applied the method of moving planes in local way to derive boundary layer estimates, then by using blowing-up arguments (see [2], [9]), they could only establish the a priori estimates for the critical order Navier problem (1.11) under the assumptions that either and Ω is strictly convex, or (see Theorem 1.3 in [4]). Now, as an immediate consequence of Theorem 1.7, we derive in Corollary 1.9 a priori estimates for the critical order Navier problem (1.11) for all the cases with no convexity assumptions on Ω, which extends Theorem 1.3 in [4] remarkably.
As a consequence of the a priori estimates (Corollary 1.9), by applying the Leray-Schauder fixed point theorem (see Theorem 4.1 in [4]), we can derive existence result for positive solution to the following Navier problem for critical order Lane-Emden equations in the full range : where is even, and is a bounded domain with boundary .
By virtue of the a priori estimates (Theorem 1.3 in [4]), using the Leray-Schauder fixed point theorem, Chen, Dai and Qin [4] obtained existence of positive solution for the critical order Navier problem (1.13) under the assumptions that either and Ω is strictly convex, or (Theorem 1.4 in [4]). For existence results on sub-critical order Hénon-Hardy equations on bounded domains, please see [17], [18], [20], [21], [26], [38] and the references therein. Since Corollary 1.9 extends Theorem 1.3 in [4] to the full range with no convexity assumptions on Ω, through entirely similar arguments, we can improve Theorem 1.4 in [4] remarkably and derive the following existence result for positive solution to the critical order Navier problem (1.13) in the full range .
Theorem 1.11 Assume . Then, the critical order Navier problem (1.13) possesses at least one positive solution . Moreover, the positive solution u satisfies
Remark 1.12 The proof of Theorem 1.11 is entirely similar to that of Theorem 1.4 in [4]. We only need to replace Theorem 1.3 in [4] by Corollary 1.9 in the proof. Thus we omit the details of the proof.
Remark 1.13 The lower bounds (1.14) on the norm of positive solutions u indicate that, if , then a uniform priori estimate does not exist and blow-up may occur when .
Consider the following integral equations associated with Navier problems for general critical order elliptic equations on : where , and the nonlinear terms .
Definition 1.14 We say that the nonlinear term f satisfies subcritical condition (on singularity), provided that is strictly increasing with respect to or for all . We say f satisfies critical condition (on singularity) provided that is invariant with respect to μ.
Remark 1.15 One should note that, being different from the subcritical growth condition on u in previous works (see e.g. [2], [4], [9], [19], [31], [34], [46]), there is no growth condition on u in Definition 1.14 of the subcritical condition (on singularity). As a consequence, with satisfies the subcritical condition (on singularity), while satisfies the critical condition (on singularity), where is arbitrary and may take various forms including: with , with and , with and , , and with , or even with , () and , and so on. The subcritical growth condition on u in previous works only admits the nonlinearity that has arbitrary order polynomial growth on u (say, with ), while the exponential type nonlinearity with satisfies the critical growth condition on u (see e.g. [2], [4], [9], [19], [31], [34], [46]).
Definition 1.16 A function defined on is called locally Lipschitz on u, provided that for any and bounded, there exists a (relatively) open neighborhood such that g is Lipschitz continuous on u in .
We need the following three assumptions on the nonlinear term .
The nonlinear term f is non-decreasing about u in , namely, There exists a such that, is locally Lipschitz on u in .
There exist a cone containing the positive -axis with vertex at 0 (for instance, ), constants , , and such that, the nonlinear term
By applying the method of scaling spheres to the generalized integral equations (1.15), we can derive the following Liouville theorem. Theorem 1.17 Assume f is subcritical (in the sense of Definition 1.14) and satisfies the assumptions , and , then the Liouville type results in Theorem 1.4 are valid for integral equations (1.15).
Remark 1.18 By using the method of scaling spheres, Theorem 1.17 can be proved through a quite similar way as in the proof of Theorem 1.4, so we leave the details to readers. We would like to mention that, if the nonlinear term satisfies subcritical conditions for (see Definition 1.14), we only need to carry out calculations and estimates outside the upper half ball during the scaling spheres procedure.
Remark 1.19 In particular, with satisfy all the assumptions in Theorem 1.17, where may take various forms including: with , with , with , and , with and , with and , or with , and , and or with , and , and so on. Thus Theorem 1.4 can be regarded as a corollary of Theorem 1.17. In addition, with and also satisfy all assumptions in Theorem 1.17.
Next, we consider the following Navier problems for general critical order equations on : where and is even. The nonlinear terms satisfy . We assume that, for any , there exist small and (may depend on x) such that .
It is clear from the proof of Theorem 1.2 that (see Section 2), under the same assumptions, the super poly-harmonic properties in Theorem 1.2 also hold for nonnegative classical solutions to the generalized critical order elliptic equations (1.19) provided that Based on the super poly-harmonic properties, one can verify under some assumptions on that the proof of Theorem 1.3 can also be adopted to show the equivalence between the generalized critical order PDEs (1.19) and IEs (1.15) (see Section 3). For these purposes, we need the following assumptions on the nonlinear term .
The nonlinear term is locally Lipschitz on u in .
There exist constants , and such that, the nonlinear term satisfies (1.20).
As a consequence of Theorem 1.17, we derive the following Liouville theorem for the generalized critical order PDEs (1.19). Theorem 1.20 Assume f is subcritical (in the sense of Definition 1.14) and satisfies the assumptions , and , then the Liouville type results in Theorem 1.6 are valid for PDEs (1.19).
Remark 1.21 In particular, with satisfies all the assumptions in Theorem 1.20, where may take various forms including: with , with and , or with and , with and , with and , and or with , and , and so on. Thus Theorem 1.6 can also be regarded as a corollary of Theorem 1.20.
Remark 1.22 When dimension , suppose satisfies the assumption with (i.e., satisfies (1.20) with and ), it is clear from the proof of Theorem 1.3 (see Section 3) that, the equivalence between the generalized critical order PDEs (1.19) and IEs (1.15) can also be proved. As a consequence of Theorem 1.17, the Liouville type results in Theorem 1.20 are also valid for and satisfies the assumption with . For instance, with , and with and , or with , and , and so on.
The rest of this paper is organized as follows. In section 2, we prove the super poly-harmonic properties for nonnegative solutions to (1.1) (i.e., Theorem 1.2) via a variant of the method used in [6]. In section 3, we show the equivalence between PDE (1.1) and IE (1.4), namely, Theorem 1.3. Section 4 is devoted to the proof of Theorem 1.4, then Theorem 1.6 follows immediately as a consequence of Theorem 1.4.
In the following, we will use C to denote a general positive constant that may depend on n, a, p and u, and whose value may differ from line to line.
Section snippets
Super poly-harmonic properties
In this section, we will prove Theorem 1.2. To this end, we make an odd extension of u to the whole space . Define where . Then u satisfies Let . We aim to show that for any and . Our proof will be divided into two steps.
Step 1. We first show that If (2.4) does not hold, then there exists , such that
Now, let
Equivalence between PDE and IE
In this section, we prove the equivalence between PDE (1.1) and IE (1.4), namely, Theorem 1.3. We only need to prove that any nonnegative solution of PDE (1.1) also satisfies IE (1.4).
(i) We first consider the cases that is even.
In section 2, we have proved that for , then (1.1) is equivalent to the following system In the following, similar as in [6], we define
The proof of Theorem 1.4
In this section, we will carry out the proof of Theorem 1.4 by applying the method of scaling spheres in integral forms developed by Dai and Qin in [21].
Suppose u is a nonnegative continuous solution of IE (1.4) but , we will derive a contradiction via the method of scaling spheres in integral forms.
In order to apply the method of scaling spheres, we first give some definitions. One can easily see that implies in . Let be an arbitrary positive real number and let the scaling
Acknowledgements
The authors are grateful to the anonymous referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.
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Wei Dai is supported by the NNSF of China (No. 11971049 and No. 11501021), the Fundamental Research Funds for the Central Universities and the State Scholarship Fund of China (No. 201806025011).