Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities ...on the entire space Rn, including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in 25, combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.
Sharp Trudinger–Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial ...differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we prove in this paper that these critical and subcritical Trudinger–Moser and Adams inequalities are actually equivalent. Moreover, we also establish the asymptotic behavior of the supremum for the subcritical Trudinger–Moser and Adams inequalities on the entire Euclidean spaces, and provide a precise relationship between the suprema for the critical and subcritical Trudinger–Moser and Adams inequalities. This relationship of supremum is useful in establishing the existence and nonexistence of extremal functions for the Trudinger–Moser inequalities. Since the critical Trudinger–Moser and Adams inequalities can be easier to prove than subcritical ones in some occasions, and more difficult to establish in other occasions, our results and the method suggest a new approach to both the critical and subcritical Trudinger–Moser and Adams type inequalities.
In this paper, we investigate the existence of nontrivial solutions to the following fractional
-Laplacian system with homogeneous nonlinearities of critical Sobolev growth:
where
denotes the ...fractional
-Laplacian operator,
,
,
,
is the critical Sobolev exponent, Ω is a bounded domain in
with Lipschitz boundary, and
and
are homogeneous functions of degrees p and q with
and
and
are the partial derivatives with respect to
and
, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional
-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional
-Laplacian equation and is of its independent interest (see Lemma
). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical
-Laplacian operators (i.e.,
) and for the single fractional
-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian
(i.e.,
and
) has not been studied in the literature before.
In this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a large class ...of parameters
and
We compute the best constants and the explicit forms of the extremal functions in numerous cases. When
, we can deduce the existence and symmetry of optimizers for a wide range of parameters. Moreover, in the particular cases
and
, the forms of maximizers will also be provided in the spirit of Del Pino and Dolbeault
,
. In the case
, that is, the Caffarelli–Kohn–Nirenberg inequality without the interpolation term, we will provide the exact maximizers for all the range of
. The Caffarelli–Kohn–Nirenberg inequalities with arbitrary norms on Euclidean spaces will also be considered in the spirit of Cordero-Erausquin, Nazaret and Villani
. Due to the absence of the classical Polyá–Szegö inequality in the weighted case, we establish a symmetrization inequality with power weights which is of independent interest.
The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the ...symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84 by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.
The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in
with infinite volume on the Sobolev-type spaces
,
, the completion of
under the norm
.... The case
(i.e.,
) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when
changes. We will study these inequalities under two types of constraint: semi-norm type
and full-norm type
,
,
. We will show that the Trudinger–Moser-type inequalities hold if and only if
. Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when
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