The existence of extremals for singular Trudinger–Moser inequalities in ℝ n involved with the trapping potential

The first purpose of this paper is to establish the singular Trudinger–Moser inequality in R 2 ${\mathbb{R}}^{2}$ for the Aharonov–Bohm magnetic fields. The second purpose is to derive the singular Hardy–Trudinger–Moser inequality in the unit ball B 2 ${\mathbb{B}}^{2}$ with Aharonov–Bohm magnetic potential. Moreover, we will show the constant 4 π ( 1 − β 2 ) $4\pi \left(1-\frac{\beta }{2}\right)$ is sharp in these two inequalities. The main skills include providing the asymptotic estimates of the related heat kernel and adapting the level set to derive a global Trudinger–Moser inequality from a local one. These results extend the inequalities established by Lu and Yang [Calc. Var. Partial Differ. Equ., 63 (2024)] into the weighted version.

In this paper, we establish the singular Trudinger–Moser inequality with norm in a bounded domain and the existence of extremal functions by blow-up analysis. As our first main result, we prove that for , , and , the singular Trudinger–Moser inequality holds, where is the first eigenvalue . Furthermore, we prove the existence of extremals for the singular Trudinger–Moser inequality with norm. Our results improve that of J. Zhu (Zhu J. Improved Moser–Trudinger inequality involving norm in n dimensions. Adv Nonlinear Stud 2014;14:273–293) into the singular case.

Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two

Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

Let be a compact Riemann surface with smooth boundary. In this paper, using blow-up analysis, we prove the existence of an extremal function for a singular Trudinger–Moser inequality on .

In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

In this paper, we study the following fractional p-Laplacian equations with exponential growth and Hardy term: where the nonlinear term satisfies exponential growth, , , λ is a positive parameter, is the fractional p-Laplacian, is a scalar potential, the weight function either falls into some Lebesgue space or is a convolution function, and g is a perturbation term. With the help of a fixed point result and singular Trudinger–Moser inequality in , we obtain the existence of a nontrivial weak solution for the above equations.

On a singular and nonhomogeneous N-Laplacian equation involving critical growth

CUDA programs for solving the time-dependent dipolar Gross–Pitaevskii equation in an anisotropic trap

In this paper, we first establish a singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any bounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular(0<β<n${(0<\beta<n}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular(β=0${(\beta=0}$) Trudinger–Moser inequality on any unbounded domain inℝn${\mathbb{R}^{n}}$with Lorentz–Sobolev norms (Theorem 1.5). The constants in all these inequalities are sharp. In [9], for the proof of Theorem 1.2 in the nonsingular caseβ=0${\beta=0}$, the following inequality was used (see [17]):u∗⁢(r)-u∗⁢(r0)≤1n⁢wn1/n⁢∫rr0U⁢(s)⁢s1/n⁢d⁢ss,$u^{\ast}(r)-u^{\ast}(r_{0})\leq\frac{1}{nw_{n}^{{1/n}}}\int_{r}^{r_{0}}U(s)s^{% {1/n}}\frac{ds}{s},$whereU⁢(x)${U(x)}$is the radial function built from|∇⁡u|${|\nabla u|}$on the level set ofu, i.e.,∫|u|>t|∇u|dx=∫0|{|u|>t}|U(s)ds.$\int_{|u|>t}\lvert\nabla u|\,dx=\int_{0}^{|\{|u|>t\}|}U(s)\,ds.$The construction of suchUuses the deep Fleming–Rishel co-area formula and the isoperimetric inequality and is highly nontrivial. Moreover, this argument will not work in the singular case0<β<n${0<\beta<n}$. One of the main novelties of this paper is that we can avoid the use of this deep construction of such a radial functionU(see remarks at the end of the introduction). Moreover, our approach adapts the symmetrization-free argument developed in [19, 21, 23], where we derive the global inequalities on unbounded domains from the local inequalities on bounded domains using the level sets of the functions under consideration.

Sequence stratigraphy is the study of rock relationships within a chronostratigraphic framework. Sequence stratigraphy is a guide to hydrocarbon prospect description and prediction. An individual sequence is a conformable succession of related strata bounded by major unconformities and corresponds to a 3rd order cycle, generally with a periodicity of a million or so years. successions of Within a sequence are parasequences, conformable related beds or bed-sets bounded by unconformities and corresponding to a 4th order cycle, with a periodicity ranging from 20 k to 100 k years. Their physical reality is based on Milankovitch climate cycles. As used here, the lateral distribution of strata and bed is global. In the northern Gulf of Mexico, an individual prospect probably formed over a period of 105 years. A hydrocarbon play, such as the Flexure Trend, evolved over a period of 10{sup 6} to 10{sup 7} years. A compilation of potential hydrocarbon trap types has been assembled for the Louisiana offshore, from coastal plain to lower slope. These potential traps are listed according to paleophysiographic provinces: coastal plain, shelf, shelf-break, upper slope, middle slope, and lower slope. Characteristics of each trap type are tabulated. The characteristics include: tectonics, regional and local sedimentation ratesmore » and types, position within an evolving sequence as determined by sequence stratigraphy, duration of reservoir and/or trap creation, and sea-level position. Regional geologic processes, such as salt tectonics, and approximate rates at which they operate are also listed. Exploration designations such as hydrocarbon province, play, and prospect may be correlated with continental margin wedge, sequence (strata), and parasequence (bed), respectively.« less

In this paper, using the method of blow-up analysis, the authors obtain an improved Trudinger–Moser inequality involving [Formula: see text]-norm ([Formula: see text]) and prove the existence of its extremal function on a closed Riemann surface [Formula: see text] with the action of a finite isometric group [Formula: see text]. To be exact, let [Formula: see text] be the usual Sobolev space, a function space [Formula: see text] and [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text] stands for the number of all distinct points in the set [Formula: see text]. Define [Formula: see text], where [Formula: see text] is the standard [Formula: see text]-norm on [Formula: see text]. Using blow-up analysis, we prove that if [Formula: see text], the supremum [Formula: see text] and this supremum can be attained; if [Formula: see text], the above supremum is infinite. This kind of inequality will play an important role in the study of prescribing Gaussian curvature problem and mean field equations. In particular, their result generalizes those of Chen [A Trudinger inequality on surfaces with conical singularities, Proc. Amer. Math. Soc. 108 (1990) 821–832], Yang [Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two, J. Differential Equations 258 (2015) 3161–3193] and Fang–Yang [Trudinger–Moser inequalities on a closed Riemannian surface with the action of a finite isometric group, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020) 1295–1324].

In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space. Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions. We also discuss the existence and non-existence of maximizers for the associated variational problem.

The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in ℝ N {\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces D N , q ⁢ ( ℝ N ) {D^{N,q}(\mathbb{R}^{N})} , q ≥ 1 {q\geq 1} , the completion of C 0 ∞ ⁢ ( ℝ N ) {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm ∥ ∇ ⁡ u ∥ N + ∥ u ∥ q {\|\nabla u\|_{N}+\|u\|_{q}} . The case q = N {q=N} (i.e., D N , q ⁢ ( ℝ N ) = W 1 , N ⁢ ( ℝ N ) {D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})} ) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type ∥ ∇ ⁡ u ∥ N ≤ 1 {\|\nabla u\|_{N}\leq 1} and full-norm type ∥ ∇ ⁡ u ∥ N a + ∥ u ∥ q b ≤ 1 {\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1} , a > 0 {a>0} , b > 0 {b>0} . We will show that the Trudinger–Moser-type inequalities hold if and only if b ≤ N {b\leq N} . 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 b > N {b>N} .

The main purpose of this paper is to establish the existence of ground-state solutions to a class of Schrödinger equations with critical exponential growth involving the nonnegative, possibly degenerate, potential V: - div ⁡ ( | ∇ ⁡ u | n - 2 ⁢ ∇ ⁡ u ) + V ⁢ ( x ) ⁢ | u | n - 2 ⁢ u = f ⁢ ( u ) . -\operatorname{div}(\lvert\nabla u\rvert^{n-2}\nabla u)+V(x)\lvert u\rvert^{n-% 2}u=f(u). To this end, we first need to prove a sharp Trudinger–Moser inequality in ℝ n {\mathbb{R}^{n}} under the constraint ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x ≤ 1 . \int_{\mathbb{R}^{n}}(\lvert\nabla u\rvert^{n}+V(x)\lvert u\rvert^{n})\,dx\leq 1. This is proved without using the technique of blow-up analysis or symmetrization argument. As far as what has been studied in the literature, having a positive lower bound has become a standard assumption on the potential V ⁢ ( x ) {V(x)} in dealing with the existence of solutions to the above Schrödinger equation. Since V ⁢ ( x ) {V(x)} is allowed to vanish on an open set in ℝ n {\mathbb{R}^{n}} , the loss of a positive lower bound of the potential V ⁢ ( x ) {V(x)} makes this problem become fairly nontrivial. Our method to prove the Trudinger–Moser inequality in ℝ 2 {\mathbb{R}^{2}} (see [L. Chen, G. Lu and M. Zhu, A critical Trudinger–Moser inequality involving a degenerate potential and nonlinear Schrödinger equations, Sci. China Math. 64 2021, 7, 1391–1410]) does not apply to this higher-dimensional case ℝ n {\mathbb{R}^{n}} for n ≥ 3 {n\geq 3} here. To obtain the existence of a ground state solution, we use a non-symmetric argument to exclude the possibilities of vanishing and dichotomy cases of the minimizing sequence in the Nehari manifold. This argument is much simpler than the one used in dimension two where we consider the nonlinear Schrödinger equation - Δ ⁢ u + V ⁢ u = f ⁢ ( u ) {-\Delta u+Vu=f(u)} with a degenerate potential V in ℝ 2 {\mathbb{R}^{2}} .