A Refined Approach for Non-Negative Entire Solutions of Δ u + u^p = 0 with Subcritical Sobolev Growth

1 Introduction

We revisit the Liouville property of non-negative entire solutions for the Lane–Emden equation

where N≥2, 1<p<(N+2)/(N-2)+ and Ω⊆ℝN. By a solution of equation (1.1), we always mean a solution in the classical sense. The Lane–Emden equation arises as a model in astrophysics and has important consequences to fundamental problems in conformal geometry. Specifically, when Ω=ℝN and p=(N+2)/(N-2), the solvability of the Lane–Emden equation is related to a sharp Sobolev inequality and the classical Yamabe problem [19]. Equation (1.1) is also the “blow-up” equation associated with a family of second-order elliptic equations with Dirichlet data. In other words, the non-existence of positive solutions to the Lane–Emden equation is important in obtaining a priori bounds to positive solutions of elliptic problems within this family [15]. The critical Sobolev exponent

plays the role of the dividing number for the existence and non-existence of solutions. More precisely, recall the following celebrated result.

The classification result of part (a) was obtained in [14] under the decay assumption that u⁢(x)=O⁢(|x|2-N) (see [26] for an equivalent result), and Caffarelli, Gidas and Spruck [3] later improved the result by dropping this decay assumption. Part (b) was obtained in [16] using local integral estimates and identities. The proofs of both parts were simplified considerably in [5] with the aid of the Kelvin transform and the method of moving planes. We refer those who are interested to [12] and the references therein for more classification and Liouville type theorems for the Lane–Emden equation with respect to various types of solutions, e.g., stable, finite Morse index, radial, sign-changing, etc.

In this paper, we use a carefully modified approach originated by Serrin and Zou [34], based on integral estimates and a Rellich–Pohozaev identity to prove the following theorem.

The initial application of Serrin and Zou’s technique was to extend this version of the Liouville type theorem to the following Lane–Emden system (p,q>0):

Obtaining an analogous non-existence theorem for this system, however, proved rather difficult. It remains an open problem and is commonly known as the Lane–Emden conjecture. More precisely, the conjecture infers that system (1.3) has no positive classical solution in Ω=ℝN if and only if the subcritical condition

holds. This subcritical condition is indeed a necessary one because of the existence result in [35], i.e., the Lane–Emden system admits a positive entire solution in the non-subcritical case. Thus, it only remains to verify the sufficiency of the subcritical condition, and this has only been achieved in special cases. The conjecture was proved under radial solutions for any dimension [24]. Then, Serrin and Zou [34] introduced their approach to prove the non-existence of positive solutions in the subcritical case but with the added restriction that solutions have polynomial growth and N=3. Poláčik, Quittner and Souplet [29] refined the method and result of [34] by removing the growth assumption on solutions, and Souplet [36] pushed it further to include the case when N=4 (for other related partial results, see [2, 6, 11, 32]). Hence, the conjecture is completely resolved for dimension N≤4. Interestingly enough, Souplet [36] further indicates that their modified approach also applies directly to equation (1.1) but in lower dimensions, of course. In view of this, our goal in the present paper is to further refine the approach to work for the scalar equation in all dimensions. We do this in order to present the underlying obstructions of the method when dealing with such elliptic equations and demonstrate how to possibly overcome them.

Another motivation for our study centers on the fact that moving plane arguments have their limitations even for related scalar problems. For example, the method of moving planes may not always apply to equations of the form

when the coefficient c⁢(x) has certain growth, e.g., c⁢(x)=|x|σ with σ>0 (see [27]). Therefore, the improved approach here may prove useful in similar situations. Despite this, the method of moving planes and its variants are powerful tools in studying properties of solutions to many elliptic equations and systems. For more details, the interested reader is referred to the following partial list of papers and the references found therein: [3, 5, 4, 13, 14, 18, 21, 23, 22, 28, 32, 33] (see also [17, 20, 25] for closely related results on Lane–Emden equations).

Roughly speaking, a key ingredient in our proof of the Liouville type theorem relies on the Rellich–Pohozaev identity. In particular, for a non-negative solution u of the subcritical Lane–Emden equation in Ω=BR⁢(0):={x∈ℝN:|x|<R} with boundary ∂⁡BR⁢(0), we have

where ν=x/|x| is the outer unit normal vector at x∈∂⁡B1⁢(0) and ∂⁡u/∂⁡ν=∇⁡u⋅ν is the directional derivative. The idea is to exploit this identity to estimate the energy quantity

Namely, the Rellich–Pohozaev identity implies the estimate

where

Here we are writing u=u⁢(r,θ) in spherical coordinates with r=|x| and x/|x|∈𝕊N-1 (x≠0), and D=Dx is the gradient operator in terms of the spatial variable x.

It will suffice to prove that equation (1.1) has no positive entire solutions. Therefore, if we assume u is indeed a positive solution, then we shall control the surface integrals above in terms of F⁢(R) to arrive at the feedback estimate F⁢(R)≤C⁢[G1⁢(R)+G2⁢(R)]≤C⁢R-a⁢F⁢(R)1-b for some a,b>0. Hence, after taking a sequence R=Rj→∞, we may conclude that ∥u∥Lp+1⁢(ℝN)=0. Thus, u≡0, and we arrive at a contradiction. The advantage of this approach is that we effectively remove one degree of freedom in the spatial dimension when estimating the energy quantity. The preceding argument, however, has a bottleneck in the sense that a weaker integrability of u is still required for the procedure to run smoothly. By weaker we mean that u is not assumed a priori to have, say, finite energy or be integrable (see Remark 2.5 below). This is due to certain standard estimates employed, e.g., the Sobolev and interpolation inequalities, which are sensitive to the dimension and are not quite sharp enough to get the feedback estimates for larger N. Specifically, for any non-negative solution u of equation (1.1), we require that there exists a number q0:=p+ε0, with

such that

where C⁢(N,p)>0 depends only on N and p. In fact, (1.6) is known to hold for any N≥2 whenever q0=p, i.e.,

This follows simply by testing equation (1.1) directly by a suitable cut-off function; see, for example, [36, 27] for the details. Then, it is clear that (1.5) is also satisfied for q0=p so long as N≤4, and this is precisely where the obstruction on the dimension appears. For N≥5, estimate (1.7) is enough to get the non-existence result but only in the non-optimal range 1<p<(N-1)/(N-3) or, equivalently,

Similar obstructions occur for the Lane–Emden system as indicated earlier in [9], which has motivated our work here. Nonetheless, we shall see below that conditions (1.5) and (1.6) are necessary and sufficient for the non-existence result.

After the completion of this paper, Professor Souplet kindly sent us his paper [37] concerning Liouville type theorems for Schrödinger elliptic systems with gradient structure, where conditions (1.5) and (1.6) also appear. There, the conditions ensured that such systems do not admit any non-negative entire solutions having certain exponential growth besides the trivial solution (see [37, Theorem 2]).

2 Proof of Theorem 1.1

We provide the details of the main steps for the reader’s convenience preferring to state without proof those intermediate results we deem standard. Meanwhile, we hope to provide enough of the details to successfully highlight where exactly the bottleneck of the approach arises. To simplify our presentation, since the main result is known in lower dimensions, we assume N≥5 unless specified otherwise.