Trudinger Moser Inequality Research Articles

New weighted Trudinger-Moser inequality for functions not necessarily radially symmetric and applications

In this paper, we prove some new inequality of Trudinger-Moser' type defined in some weighted Sobolev space whose norm is a combination of the norms of the N-Laplacian and the p-Laplacian with 1<p<N in RN,N≥2. The presence and the behavior of the weights prevent us to use of the symmetrization arguments. This inequality holds for functions which are not required to be radially symmetric. The strategy used in order to establish our inequality mainly consists of a proper change of variables which permits to return to the non-weighted case. Various improvements of this inequality are also proved. Finally, one of these improvements is used to study some elliptic equation involving a weighted (N,p)-Laplacian operator.

On a Schrödinger equation involving fractional [formula omitted]-Laplacian with critical growth and Trudinger–Moser nonlinearity

A nonlinear Schrödinger equation of fractional (N/s1,q)-Laplacian is considered with the Rabinowitz potential, critical Sobolev growth and Trudinger–Moser nonlinearity in RN−ΔN/s1s1u+−Δqs2u+V(ɛx)(uNs1−2u+uq−2u)=λfu+uqs2∗−2u.We establish the global existence of nonnegative ground-state solution for suitable parameter values primarily through variational analysis, fractional Trudinger–Moser inequality and mountain pass approach. It is a crucial ingredient to handle three aspects concerning the limiting setting s1p=N, the critical Sobolev growth and Trudinger–Moser nonlinearity.

Magnetic Kirchhoff–Schrödinger equations with critical exponential growth and indefinite potential in R 2

In this paper, we are concerned with the following magnetic nonlinear equation of Kirchhoff type with critical exponential growth and an indefinite potential in R 2 m ( ∫ R 2 [ | 1 i ∇ u − A ( x ) u | 2 + V ( x ) | u | 2 ] d x ) [ ( 1 i ∇ − A ( x ) ) 2 u + V ( x ) u ] = B ( x ) f ( | u | 2 ) u , where u ∈ H 1 ( R 2 , C ), m is a Kirchhoff type function, V : R 2 → R and A : R 2 → R 2 represent locally bounded potentials, while B denotes locally bounded and f exhibits critical exponential growth. By employing variational methods and utilizing the modified Trudinger–Moser inequality, we get ground state solutions or nontrivial solutions for the above equation. Furthermore, in the special case where m is a constant equal to 1, the equation is reduced to the following magnetic nonlinear Schrödinger equation, ( 1 i ∇ − A ( x ) ) 2 u + V ( x ) u = B ( x ) f ( | u | 2 ) u in R 2 . Applying analogous methods, we can also establish the existence of ground state solutions or nontrivial solutions to this equation.

Moser–Trudinger inequalities: from local to global

Moser–Trudinger inequalities: from local to global

Extremal Functions for Trudinger–Moser Inequalities Involving Various $L^{p}$-norms in High Dimension

Extremal Functions for Trudinger–Moser Inequalities Involving Various $L^{p}$-norms in High Dimension

Extremals of singular Hardy–Trudinger–Moser inequality with remainder terms on unit disc

Extremals of singular Hardy–Trudinger–Moser inequality with remainder terms on unit disc

Planar Schrödinger-Poisson system with zero mass potential and critical exponential growth

This paper is concerned with the following planar Schrödinger-Poisson system with zero mass potential{−Δu+ϕu=Q(|x|)f(u),x∈R2,Δϕ=u2,x∈R2, where Q∈C((0,∞),(0,∞)) may be singular or unbound and f∈C(R,R) is of critical exponential growth and there is no monotonicity restriction on f(u)/u3. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality within the working space associated with the energy functional relevant to the aforementioned problem, which is different from Albuquerque et al. (2021) [5]. By combining the variational methods and delicate estimates, we prove the existence of a non-trivial mountain-pass solution to the above system under mild assumptions on Q and f.

Trudinger–Moser Inequalities on a Closed Riemann Surface with a Symmetric Conical Metric

Trudinger–Moser Inequalities on a Closed Riemann Surface with a Symmetric Conical Metric

Positive solution for a singular and nonlocal problem of the N-Kirchhoff type

We investigate the existence of a positive solution for a singular elliptic problem of the N-Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation consists of two terms: one has a subcritical, critical or supercritical exponential growth governed by the Trudinger–Moser inequality, and the other one presents a singularity at the origin. We also show a result of nonexistence of solutions in dimension 2. Finally, we study the asymptotic behavior of the solutions with respect to the parameter.

Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities

Abstract In this article, we consider the existence of solutions for nonlinear elliptic equations of the form − Δ u + V ( ∣ x ∣ ) u = Q ( ∣ x ∣ ) f ( u ) , x ∈ R 2 , -\Delta u+V\left(| x| )u=Q\left(| x| )f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{2}, where the nonlinear term f ( s ) f\left(s) has critical exponential growth which behaves like e α s 2 {e}^{\alpha {s}^{2}} , the radial potentials V , Q : R + → R V,Q:{{\mathbb{R}}}^{+}\to {\mathbb{R}} are unbounded, singular at the origin or decaying to zero at infinity. By combining the variational methods, Trudinger-Moser inequality, and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a Mountain-pass-type solution for the above problem under some weak assumptions.

Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials

Abstract The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x &lt; + ∞ , $$\underset{u\in {H}^{1}\left({\mathbb{R}}^{2}\right),\underset{{\mathbb{R}}^{2}}{\int }\left(\vert \nabla u{\vert }^{2}+V\left(x\right){u}^{2}\right)\mathrm{d}x\le 1}{\mathrm{sup}}\underset{{\mathbb{R}}^{2}}{\int }\frac{{e}^{4\pi \left(1-\frac{\beta }{2}\right){u}^{2}}-1}{\vert x{\vert }^{\beta }}\mathrm{d}x{&lt; }+\infty ,$$ where 0 &lt; β &lt; 2 $0{&lt; }\beta {&lt; }2$ , V ( x ) ≥ 0 $V\left(x\right)\ge 0$ and may vanish on an open set in R 2 ${\mathbb{R}}^{2}$ . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 ${\mathbb{R}}^{2}$ : (0.2) − Δ u + γ u = f ( u ) | x | β , $$-{\Delta}u+\gamma u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }},$$ where the nonlinearity f $f$ has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, “Concentration-compactness principle of singular Trudinger-Moser inequalities in R n ${\mathbb{R}}^{n}$ and n $n$ -Laplace equations,” Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 ${\mathbb{R}}^{2}$ : (0.3) − Δ u + V ( x ) u = f ( u ) | x | β . $$-{\Delta}u+V\left(x\right)u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }}.$$

Trudinger–Moser and Hardy–Trudinger–Moser inequalities for the Aharonov–Bohm magnetic field

Trudinger–Moser and Hardy–Trudinger–Moser inequalities for the Aharonov–Bohm magnetic field

Sharp anisotropic singular Trudinger–Moser inequalities in the entire space

Sharp anisotropic singular Trudinger–Moser inequalities in the entire space

Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N

Abstract 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.

A planar Kirchhoff equation with exponential growth and double nonlocal term

We investigate the existence of a positive solution for an elliptic problem of the Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation combined a nonlocal term with an exponential growth governed by the Trudinger-Moser inequality.

Semiclassical Moser–Trudinger inequalities

We extend the Moser–Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schrödinger operators on bounded domains.

Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight

Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight ωβ(x)=[−lnFo(x)|(n−1)β. This can be seen as a generation result of the isotropic Moser–Trudinger inequality with logarithmic weight. Furthermore, we obtain the existence of extremal function when β is small. Finally, we give Lions’ concentration-compactness principle, which is the improvement of the anisotropic Moser–Trudinger-type inequality.

Existence of solutions to quasilinear Schrodinger equations with exponential nonlinearity

In this article we study the existence of solutions to quasilinear Schrodinger equations in the plane, involving a potential that can change sign and a nonlinear term that may be discontinuous and exhibit exponential critical growth. To prove our existence result, we combine the Trudinger-Moser inequality with a fixed point theorem. For mote information see https://ejde.math.txstate.edu/Volumes/2024/14/abstr.html

Sign-changing solutions for a weighted Kirchhoff problem with exponential growth non-linearity

In this work, we establish the existence of solutions that change sign at low energy for a non-local weighted Kirchhoff problem in the set }2 $ ]]> R N , N > 2 . The non-linearity of the equation is assumed to have exponential growth in view of the logarithmic weighted Trudinger–Moser inequalities. To obtain the existence result, we apply the constrained minimization in the Nehari set, the quantitative deformation lemma and results from degree theory.

The fully parabolic multi-species chemotaxis system in

Abstract This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$ . The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.