Trudinger Moser Inequality Research Articles

On a weighted elliptic equation of N-Kirchhoff type with double exponential growth

Abstract In this work, we study the weighted Kirchhoff problem − g ∫ B σ ( x ) ∣ ∇ u ∣ N d x div ( σ ( x ) ∣ ∇ u ∣ N − 2 ∇ u ) = f ( x , u ) in B , u > 0 in B , u = 0 on ∂ B , \left\{\begin{array}{ll}-g\left(\mathop{\displaystyle \int }\limits_{B}\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N}{\rm{d}}x\right){\rm{div}}\left(\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N-2}\nabla u)=f\left(x,u)& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial B,\end{array}\right. where B B is the unit ball of R N {{\mathbb{R}}}^{N} , σ ( x ) = log e ∣ x ∣ N − 1 \sigma \left(x)={\left(\log \left(\frac{e}{| x| }\right)\right)}^{N-1} , the singular logarithm weight in the Trudinger-Moser embedding, and g g is a continuous positive function on R + {{\mathbb{R}}}^{+} . The nonlinearity is critical or subcritical growth in view of Trudinger-Moser inequalities. We first obtain the existence of a solution in the subcritical exponential growth case with positive energy by using minimax techniques combined with the Trudinger-Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth, and we stress its importance to check the compactness level.

On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth

Abstract In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1) − Δ u + V ( x ) u = f ( x , u ) + λ a ( x ) ∣ u ∣ q − 2 u , x ∈ R 2 , -\Delta u+V\left(x)u=f\left(x,u)+\lambda a\left(x)| u{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2}, where λ > 0 \lambda \gt 0 , q ∈ ( 1 , 2 ) q\in \left(1,2) , a ∈ L 2 / ( 2 − q ) ( R 2 ) a\in {L}^{2\text{/}\left(2-q)}\left({{\mathbb{R}}}^{2}) , V ( x ) V\left(x) , and f ( x , t ) f\left(x,t) are 1-periodic with respect to x x , and f ( x , t ) f\left(x,t) has critical exponential growth at t = ∞ t=\infty . By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. λ > 0 \lambda \gt 0 ) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.

On concentration of solutions for quasilinear Schrödinger equations with critical growth in the plane

In this work, we study the existence of positive ground states and concentration phenomena for the following class of quasilinear Schrödinger equations: where is a parameter, is a continuously differentiable function, and are positive and bounded continuous potentials and the nonlinearity can exhibit critical exponential growth. In order to prove our concentration result, we exploit variational arguments that take into account an interaction between the potentials V and Q, in combination with the Trudinger–Moser inequality and Moser iteration method.

Normalized solutions of Chern-Simons-Schrödinger equations with exponential critical growth

This paper is concerned with the existence of normalized solutions to the following Chern-Simons-Schrödinger equation with an exponential critical reaction:{−Δu−λu+(h2(|x|)|x|2+∫|x|+∞h(s)su2(s)ds)u=K(|x|)f(u)inR2,∫R2|u|2dx=c2, where h(s)=∫0sl2u2(l)dl, λ∈R is unknown and appears as a Lagrange multiplier, K∈C1(R2,R+) is the radially symmetric potential and f has exponential critical growth in the sense of Trudinger-Moser inequality. By the minimization approach in combination with some new tricks, we overcome the difficulties resulting from the non-constant potential K(|x|) and the exponential-critical-growth reaction. The main results establish the existence of normalized solutions to the above Chern-Simons-Schrödinger equation, as well as in the case K(|x|)=1.

Aubin Type Almost sharp Moser–Trudinger Inequality Revisited

We give a new proof of the almost sharp Moser–Trudinger inequality on smooth compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces and generalize it to Riemannian manifolds with continuous metrics and to higher order Sobolev spaces on manifolds with boundary under several boundary conditions. These generalizations can be applied to fourth order Q curvature equations in dimension 4.

Trudinger–Moser type inequalities with a symmetric conical metric and a symmetric potential

Let (M,g) be a closed Riemann surface, where the metric g has certain conical singularities at finite points. Suppose Γ is a group with elements of isometries acting on (M,g). In this paper, Trudinger–Moser inequalities involving Γ and the operador Δg+V are established, where Δg denotes the Laplace–Beltrami operator associated to g and the potential V:M→(0,∞) belongs to a class of symmetric and continuous functions. Moreover, via the method of blow-up analysis, the corresponding extremals are also obtained.

Equivalence of critical and subcritical sharp Trudinger–Moser inequalities in fractional dimensions and extremal functions

We establish critical and subcritical sharp Trudinger–Moser inequalities for fractional dimensions on the whole space. Moreover, we obtain asymptotic lower and upper bounds for the fractional subcritical Trudinger–Moser supremum from which we can prove the equivalence between critical and subcritical inequalities. Using this equivalence, we prove the existence of maximizers for both the subcritical and critical associated extremal problems. As a by-product of this development, we can explicitly calculate the value of the critical supremum in some special situations.

Planar Schrödinger-Choquard equations with potentials vanishing at infinity: The critical case

We study the following class of stationary Schrödinger equations of Choquard type−Δu+V(x)u=[|x|−μ⁎(Q(x)F(u))]Q(x)f(u),x∈R2, where the potential V and the weight Q decay to zero at infinity like (1+|x|γ)−1 and (1+|x|β)−1 for some (γ,β) in variously different ranges, ⁎ denotes the convolution operator with μ∈(0,2), and F is the primitive of f that fulfills a critical exponential growth in the Trudinger-Moser sense. By establishing a version of the weighted Trudinger-Moser inequality, we investigate the existence of nontrivial solutions of mountain-pass type for the given problem. Furthermore, we shall establish that the nontrivial solution is a bound state, namely a solution belonging to H1(R2), for some particular (γ,β).

On the planar Kirchhoff-type problem involving supercritical exponential growth

Abstract This article is concerned with the following nonlinear supercritical elliptic problem: − M ( ‖ ∇ u ‖ 2 2 ) Δ u = f ( x , u ) , in B 1 ( 0 ) , u = 0 , on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-M(\Vert \nabla u{\Vert }_{2}^{2})\Delta u=f\left(x,u),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R 2 {{\mathbb{R}}}^{2} , M : R + → R + M:{{\mathbb{R}}}^{+}\to {{\mathbb{R}}}^{+} is a Kirchhoff function, and f ( x , t ) f\left(x,t) has supercritical exponential growth on t t , which behaves as exp [ ( β 0 + ∣ x ∣ α ) t 2 ] \exp {[}({\beta }_{0}+| x\hspace{-0.25em}{| }^{\alpha }){t}^{2}] and exp ( β 0 t 2 + ∣ x ∣ α ) \exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }}) with β 0 {\beta }_{0} , α > 0 \alpha \gt 0 . Based on a deep analysis and some detailed estimate, we obtain Nehari-type ground state solutions for the above problem by variational method. Moreover, we can determine a fine upper bound for the minimax level under weaker assumption on liminf t → ∞ t f ( x , t ) exp [ ( β 0 + ∣ x ∣ α ) t 2 ] {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp {[}({\beta }_{0}+| \hspace{-0.25em}x\hspace{-0.25em}{| }^{\alpha }){t}^{2}]} and liminf t → ∞ t f ( x , t ) exp ( β 0 t 2 + ∣ x ∣ α ) {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }})} , respectively. Our results generalize and improve the ones in G. M. Figueiredo and U. B. Severo (Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), no. 1, 23–39.) and Q. A. Ngó and V. H. Nguyen (Supercritical Moser-Trudinger inequalities and related elliptic problems, Calc. Var. Partial Differ. Equ. 59 (2020), no. 2, Paper No. 69, 30.) for M ( t ) = 1 M(t)=1 . In particular, if the weighted term ∣ x ∣ α | x\hspace{-0.25em}{| }^{\alpha } is vanishing, we can obtain the ones in S. T. Chen, X. H. Tang, and J. Y. Wei (2021) (Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), no. 1, Paper No. 38, Theorem 1.3 and Theorem 1.4) immediately.

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

Schrödinger equations in [formula omitted] with critical exponential growth and concave nonlinearities

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form(0.1)−Δu+V(|x|)u=Q(|x|)f(u)+λg(|x|,u),x∈R2, where the nonlinear term f(s) has critical exponential growth which behaves like eαs2, g(r,s) is a concave term on s, the radial potentials V,Q:R+→R are unbounded, singular at the origin or decaying to zero at infinity and λ>0 is a parameter. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. 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 nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.

Nonexistence of Extremals for a Trudinger–Moser Inequality on a Riemann Surface with Boundary

Nonexistence of Extremals for a Trudinger–Moser Inequality on a Riemann Surface with Boundary

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

In this paper, we prove the following nonlocal Kirchhoff problem of the type has a Nehari‐type ground state solution when and are periodic on and has critical exponential growth in the sense of Trudinger–Moser inequality on . We develop some new approaches to estimate precisely the minimax level of the energy functional and to recover the compactness of Cerami sequences of the associated Euler–Lagrange functional. With this in hand, we can overcome difficulties arising from the appearance of the nonlocal term and the nonlinearity which is of critical growth of Trudinger–Moser type in whole Euclidean space .

Improved Trudinger–Moser inequality involving Lp-norm on a closed Riemann surface with isometric group actions

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].

Gagliardo-Nirenberg type inequalities on Lorentz, Marcinkiewicz and weak-𝐿^{∞} spaces

We establish the Gagliardo-Nirenberg inequality, Trudinger-Moser inequality and John-Nirenberg inequality using the Lorentz spaces L p , α L^{p,\alpha } , the Marcinkiewicz space L q , ∞ L^{q,\infty } and the weak- L ∞ L^{\infty } space W W introduced by Bennett, DeVore and Sharpley [Ann. of Math. (2) 113 (1981), pp. 601–611]. As consequences, we obtain the Gagliardo-Nirenberg type inequality with weak- L ∞ L^{\infty } norm and BMO norm, Trudinger-Moser type inequality and John-Nirenberg type estimate with B M O BMO norm and weak- L 1 L^{1} norm.

Existence and multiplicity of solutions to N-Kirchhoff equations with critical exponential growth and a perturbation term

The aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: where Ω is a smooth bounded domain in and . Secondly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: where and . We assume that f and g have critical exponential growth at infinity. To establish our existence results, we use the mountain pass theorem, Ekeland variational principle and Moser–Trudinger inequality.

Stability for the wave equation in an unbounded domain with finite measure and with nonlinearities of arbitrary growth

In this paper we study the stability of the energy associated to an initial boundary value problem involving a semilinear wave equation. The domain is an unbounded subset of R2 with unbounded boundary. On the function responsible by the semilinearity we consider a very general decay rate control. The main tools are the Trudinger–Moser inequality combined with microlocal analysis arguments and the techniques of Lasiecka and Tataru.

Soliton solutions for a class of Schrödinger equations with a positive quasilinear term and critical growth

AbstractWe consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics $-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$ in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms $h(s)$ with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$ when the parameter $\kappa >0$ is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.

A weighted Trudinger–Moser inequality on a closed Riemann surface with a finite isometric group action

A weighted Trudinger–Moser inequality on a closed Riemann surface with a finite isometric group action

Mountain-pass type solutions for the Chern-Simons-Schrödinger equation with critical exponential growth

In this paper, we consider the nonlinear Chern-Simons-Schrödinger equations where is the so-called Chern-Simons term, has critical exponential growth which behaves like . By combing 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. Our results improve and extend the previous results.