A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems
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 into the Lebesgue spaces with , where with is a bounded domain. If working in a larger class of “rearrangement invariant” Banach spaces rather than the class of -spaces, the optimal exponent can be slightly improved. For example, the following embedding is well-known where 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 as target spaces, where B is the unit ball in , the authors in [6] are able to go beyond the critical threshold when restricting to the first order Sobolev space of radially symmetric functions about the origin. This special space is simply the completion of under the norm where we denote by the class of compactly supported, smooth, radially symmetric functions about the origin in B. The primary result in [6] states that given there exists a positive constant such that the supercritical Sobolev inequality holds for any . In other words, there is a continuous embedding where is the variable exponent Lebesgue space defined by with norm As an application of (1.1), which is quite a surprise, the authors are able to prove that the following elliptic equation admits at least one solution. This result is somewhat intriguing because if one replaces 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 embedding where the space 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
To state our results, several notations and conventions are needed. First, for an integer , we denote By we mean the usual Sobolev space on B, which is the completion of under the norm Then, in analogy to , we denote by the completion of with respect to the preceding norm. Given , we are interested in whether or not the following inequality holds for some constant . Note that, in (1.4) and under the condition , the number is also the critical exponent for the following Sobolev inequality with the sharp constant for any . It is well-known that the sharp constant can be characterized by (see the formulas (2.9) and (2.10) below) and if we let then we immediately have The first main result in this paper answers the above question affirmatively.
Theorem 1.1 Let and . Then
Apparently, as the case was already studied in [6], our contribution is for the case . Clearly, a consequence of Theorem 1.1 is that the space can be continuously embedded into the variable exponent Lebesgue space mentioned earlier. An exact statement of this fact is as follows:
Corollary 1.2 Let and . Then the following embedding is continuous where is the variable exponent Lebesgue space defined by with norm
In view of Theorem 1.1, there exists a sharp constant as already given in (1.4). In this sense, it is natural to ask whether or not the sharp constant is attained. To obtain the attainability of the sharp constant and inspired by [6, Theorem 1.3], we first establish certain estimates between and as shown in the following.
Theorem 1.3 Let and . Then, there always holds Moreover, if then there holds Finally, the following limit holds.
In view of (1.6), it is now clear to see how reasonable the condition appearing in [6, Theorem 1.4] is. Compared to [6, Theorem 1.3], it is clear that, even when , 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 is attained.
Theorem 1.4 Let and . If then the sharp constant is attained.
Combining Theorem 1.3, Theorem 1.4 we deduce that the sharp constant is attained if and it is likely that the sharp constant serves as a threshold for the existence of optimizers for . Although we cannot say anything about the inequality (1.8) whenever , the limit in (1.9) might lead us to a non-existence of optimizers for when α is very large. If this is not the case, we expect to see certain monotonicity of 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 -type perturbation; see [8].
Finally, we study the existence of solutions to (1.3). Our existence result reads as follows.
Theorem 1.5 Let and . 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 . 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 : proof of Theorem 1.1
Instead of proving Theorem 1.1 for the exponent , we shall prove a more general result for the exponent where f is a function satisfying the assumptions () and () from Lemma 2.3. In this sense, we are about to show that Furthermore, because the case was already considered in [6], we do not treat the case here. Instead, we only consider the case . Let such that . 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, Since the above problem has a variational structure, we employ variational methods. To this purpose, we consider the functional on . By Theorem 1.1, the functional I is well-defined and of class on . Consequently, if is a critical point of I, namely,
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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Funded by the Simons Foundation Grant Targeted for Institute of Mathematics, Vienam Academy of Science and Technology.