A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

Dedicated to Professor Nguyễn Vǎn Mậu on the occasion of his 70th birthday
https://doi.org/10.1016/j.jde.2019.11.014Get rights and content

Abstract

A Sobolev type embedding for radially symmetric functions on the unit ball B in Rn, n3, into the variable exponent Lebesgue space L2+|x|α(B), 2=2n/(n2), α>0, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequalitysup{B|u(x)|2+|x|αdx:uH0,rad1(B),uL2(B)=1}<+ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on B, namely, the embeddingH0,radm(B)L2m+|x|α(B) with 2m<n/2, 2m=2n/(n2m), and α>0 holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime.

Introduction

The Sobolev embedding is a basic tool in many aspects of mathematical analysis. The classical one provides an optimal embedding from the Sobolev space H1(Ω) into the Lebesgue spaces Lp(Ω) with p2:=2n/(n2), where ΩRn with n3 is a bounded domain. If working in a larger class of “rearrangement invariant” Banach spaces rather than the class of Lp-spaces, the optimal exponent 2 can be slightly improved. For example, the following embedding is well-knownH1(Ω)L2,2(Ω), where L2,2(Ω) is the well-known Lorentz space. In the literature, Sobolev embedding into non-rearrangement invariant spaces has recently captured attention. By choosing the variable exponent Lebesgue spaces Lp(x)(B) as target spaces, where B is the unit ball in Rn, the authors in [6] are able to go beyond the critical threshold 2 when restricting to H0,rad1(B) the first order Sobolev space of radially symmetric functions about the origin. This special space is simply the completion of C0,rad(B) under the normuH0,rad1(B)=(B|u|2dx)1/2, where we denote by C0,rad(B) the class of compactly supported, smooth, radially symmetric functions about the origin in B. The primary result in [6] states that given α>0 there exists a positive constant Un,α such that the supercritical Sobolev inequalityUn,α:=sup{B|u(x)|2+|x|αdx:uH0,rad1(B),uL2(B)=1}<+ holds for any uH0,rad1(B). In other words, there is a continuous embeddingH0,rad1(B)L2+|x|α(B), where L2+|x|α is the variable exponent Lebesgue space defined byL2+|x|α(B):={u:BRis measurable:B|u(x)|2+|x|αdx<+} with normuL2+|x|α(B)=inf{λ>0:B|u(x)λ|2+|x|αdx1}. As an application of (1.1), which is quite a surprise, the authors are able to prove that the following elliptic equation{Δu=u2+|x|α1 in B,u>0 in B,u=0 on B, admits at least one solution. This result is somewhat intriguing because if one replaces |x|α by any non-negative constant, then (1.2) has no solution by the classical result of Pohozaev. The above new Sobolev type embedding is remarkable and applications to this embedding is now taking shape; see [3], [4], [5].

In this work, motivated by the supercritical Sobolev inequality (1.1), first we generalize (1.1) for higher order Sobolev space of radially symmetric functions leading us to the following continuous embeddingH0,radm(B)L2m+|x|α(B), where the space L2m+|x|α(B) is precisely mentioned in Corollary 1.2 below. Then as an application of the inequality we present an existence result for solutions to the following polyharmonic equation{(Δ)mu=u2m+|x|α1 in B,u>0 in B,rju=0 on B,j=0,,m1.

To state our results, several notations and conventions are needed. First, for an integer m1, we denotem={Δm/2ifmis even,Δ(m1)/2ifmis odd. By H0m(B) we mean the usual Sobolev space on B, which is the completion of C0(B) under the normuH0m(B)=(B|mu|2dx)1/2. Then, in analogy to H0,rad1(B), we denote by H0,radm(B) the completion of C0,rad(B) with respect to the preceding norm. Given α>0, we are interested in whether or not the following inequalityUn,m,α:=sup{B|u(x)|2m+|x|αdx:uH0,radm(B),muL2(B)1}<+ holds for some constant Un,m,α>0. Note that, in (1.4) and under the condition n>2m, the number 2m=2n/(n2m) is also the critical exponent for the following Sobolev inequality with the sharp constant Sn,muL2m(B)Sn,mmuL2(B) for any uH0m(B). It is well-known that the sharp constant Sn,m can be characterized bySn,m=sup{uL2m(B):uH0,radm(B),muL2(B)=1} (see the formulas (2.9) and (2.10) below) and if we letΣn,m:=sup{B|u(x)|2mdx:uH0,radm(B),muL2(B)=1}, then we immediately haveΣn,m=Sn,m2m. The first main result in this paper answers the above question affirmatively.

Theorem 1.1

Let 1m<n/2 and α>0. Thensup{B|u(x)|2m+|x|αdx:uH0,radm(B),muL2(B)1}<+.

Apparently, as the case m=1 was already studied in [6], our contribution is for the case m2. Clearly, a consequence of Theorem 1.1 is that the space H0,radm(B) can be continuously embedded into the variable exponent Lebesgue space L2m+|x|α(B) mentioned earlier. An exact statement of this fact is as follows:

Corollary 1.2

Let 1m<n/2 and α>0. Then the following embedding is continuousH0,radm(B)L2m+|x|α(B), where L2m+|x|α is the variable exponent Lebesgue space defined byL2m+|x|α(B):={u:BRis measurable:B|u(x)|2m+|x|αdx<+} with normuL2m+|x|α(B)=inf{λ>0:B|u(x)λ|2m+|x|αdx1}.

In view of Theorem 1.1, there exists a sharp constant Un,m,α>0 as already given in (1.4). In this sense, it is natural to ask whether or not the sharp constant Un,m,α is attained. To obtain the attainability of the sharp constant Un,m,α and inspired by [6, Theorem 1.3], we first establish certain estimates between Un,m,α and Σn,m as shown in the following.

Theorem 1.3

Let 1m<n/2 and α>0. Then, there always holdsUn,m,αΣn,m. Moreover, if0<αn2m, then there holdsUn,m,α>Σn,m. Finally, the following limitlimα+Un,m,α=Σn,m holds.

In view of (1.6), it is now clear to see how reasonable the condition Un>Σn appearing in [6, Theorem 1.4] is. Compared to [6, Theorem 1.3], it is clear that, even when m=1, which was also studied in [6], the range for α in (1.7) is significantly improved.

Then the following result provides us a criteria in which the sharp constant Un,m,α is attained.

Theorem 1.4

Let 1m<n/2 and α>0. IfUn,m,α>Σn,m, then the sharp constant Un,m,α is attained.

Combining Theorem 1.3, Theorem 1.4 we deduce that the sharp constant Un,m,α is attained if 0<αn2m and it is likely that the sharp constant Σn,m serves as a threshold for the existence of optimizers for Un,m,α. Although we cannot say anything about the inequality (1.8) whenever α>n2m, the limit in (1.9) might lead us to a non-existence of optimizers for Un,m,α when α is very large. If this is not the case, we expect to see certain monotonicity of Un,m,α with respect to α; see [11] for related results. We take this chance to mention that in the literature a similar phenomenon appears in the Adimurthi–Druet inequality, an improvement of the standard Moser–Trudinger inequality by adding a L2-type perturbation; see [8].

Finally, we study the existence of solutions to (1.3). Our existence result reads as follows.

Theorem 1.5

Let 1m<n/2 and 0<αn2m. Then there exists at least one weak solution to (1.3).

To look for a solution to (1.3), we employ variational techniques. In this way, a solution to (1.3) is found as a critical point of the associated Euler–Lagrange energy functional defined on H0,radm(B). In turn, such a solution is radially symmetric. Taking the recent work [3] into account, we expect to see more solution to (1.3) instead of the radial ones.

This is the first paper in a set of our works concerning functional inequalities in the supercritical regime. In the next paper [9], we shall address supercritical Moser–Trudinger inequalities and related elliptic problems.

Section snippets

Preliminaries

This section is to prepare some auxiliary results which will be used in the proof of the main Theorems.

The existence of the sharp constant Un,m,α: proof of Theorem 1.1

Instead of proving Theorem 1.1 for the exponent 2m+|x|α, we shall prove a more general result for the exponent2m+f(|x|), where f is a function satisfying the assumptions (f1) and (f2) from Lemma 2.3. In this sense, we are about to show thatsup{B|u(x)|2m+f(|x|)dx:uH0,radm(B),muL2(B)1}<+. Furthermore, because the case m=1 was already considered in [6], we do not treat the case m=1 here. Instead, we only consider the case m2. Let uC0,rad(B) such that muL2(Ω)1. The Hardy–Rellich

Higher order elliptic problems: proof of Theorem 1.5

We now turn our attention to the existence result for solutions to (1.3), namely,{(Δ)mu=u2m+|x|α1 in B,u>0 in B,rju=0 on B,j=0,,m1. Since the above problem has a variational structure, we employ variational methods. To this purpose, we consider the functionalI(u)=12B|mu|2dxB12m+|x|αu+(x)2m+|x|αdx on H0,radm(B). By Theorem 1.1, the functional I is well-defined and of class C1 on H0,radm(B). Consequently, if uH0,radm(B) is a critical point of I, namely,I(u),ϕ=Bmu,mϕdxB(u+)2m

Acknowledgements

The authors would like to thank an anonymous referee for careful reading of the paper and valuable comments, which helps us to improve the paper. The research of Q.A.N. is funded by the Vietnam National University, Hanoi (VNU) under project number QG.19.12. The research of V.H.N. is funded by the Simons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science and Technology.

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