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In the present paper, we establish an infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with double normal frequencies. Applying it, we prove the existence of quasi-periodic solutions for one dimensional coupled nonlinear quantum harmonic oscillators (QHO) with a natural reversible structure. To compensate the lack of smoothing effect of perturbation, we introduce a class of vector fields with polynomial decay which extends the works of Grébert and Thomann (Commun Math Phys 307(2):383–427, 2011) for Hamiltonian QHO. To deal with the reversible, coupled perturbations in the equations, we also introduce a new class of generating vector fields during the KAM iteration. Moreover, the quasi-periodic solutions we obtain may not be linearly stable. This is obviously different from the result in Grébert and Thomann (2011) for Hamiltonian QHO. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the following fourth order Schrödinger equation with mixed dispersion on M N , an N-dimensional Cartan-Hadamard manifold. Namely we consider 4NLS i ∂ t ψ = - Δ M 2 ψ + β Δ M ψ + λ | ψ | 2 σ ψ in R × M , ψ (0 , ·) = ψ 0 ∈ X , where β ≥ 0 , λ = { - 1 , 1 } , 0 < σ < 4 / (N - 4) + , Δ M is the Laplace-Beltrami operator on M and X = L 2 (M) or X = H 2 (M) . At first, we focus on the case where M is the hyperbolic space H N . Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to (4NLS) as well as scattering for small initial data provided that N ≥ 4 and 0 < σ < 4 / N if X = L 2 (H N) or 0 < σ < 4 / (N - 4) + if X = H 2 (H N) . Next, we obtained weighted Strichartz estimates for radial solutions to (4NLS) on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument. [ABSTRACT FROM AUTHOR]

We study the initial-boundary vaule problem of the long and short wave equations posed on the half line with initial datas (u 0 , n 0) ∈ H s 0 (R +) × H s 1 (R +) and boundary datas (h , f) ∈ H 2 s 0 + 1 4 (R +) × H s 1 (R +) . We show the local well-posedness by giving the bilinear estimates of the coupling terms for the equations in suitable spaces of Bourgain type. Moreover, we consider results concerning ill-posedness for the system. Finally, the system is proved to be globally well-posed in Sobolev spaces H 1 (R +) × L 2 (R +) . [ABSTRACT FROM AUTHOR]

In this paper, we consider the inhomogeneous nonlinear Schrödinger equation i ∂ t u + Δ u = K (x) | u | α u , u (0) = u 0 ∈ H 1 (R N) , N ≥ 3 , | K (x) | + | x | | ∇ K (x) | ≲ | x | - b , 0 < b < min (2 , N - 2) , 0 < α < (4 - 2 b) / (N - 2) . We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted L 2 -space for a new range α 0 (b) < α < (4 - 2 b) / N . The value α 0 (b) is the positive root of N α 2 + (N - 2 + 2 b) α - 4 + 2 b = 0 , which extends the Strauss exponent known for b = 0 . Our results improve the known ones for K (x) = μ | x | - b , μ ∈ C . For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of α . In the defocusing case, we prove decay estimates provided that the potential satisfies some rigidity-type condition which leads to a scattering result. We give also a new scattering criterion taking into account the potential K. [ABSTRACT FROM AUTHOR]

We derive a family of interpolation estimates which improve Hardy's inequality and cover the Sobolev critical exponent. We also determine all optimizers among radial functions in the endpoint case and discuss open questions on nonrestricted optimizers. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension. [ABSTRACT FROM AUTHOR]

In this paper we consider the following focusing mass-critical Hartree equation with a defocusing perturbation and harmonic potential i ∂ t ψ = - Δ ψ + | x | 2 ψ - (| x | - 2 ∗ | ψ | 2 ) ψ + ε | ψ | p - 2 ψ , in R + × R N , where N ≥ 3 , 2 < p < 2 ∗ = 2 N / (N - 2) and ε > 0 . We mainly focus on the normalized ground state solitary waves of the form ψ (t , x) = e i μ t u ε , ρ (x) , where u ε , ρ (x) is radially symmetric-decreasing and ∫ R N | u ε , ρ | 2 d x = ρ . Firstly, we prove the existence and nonexistence of normalized ground states under the L 2 -subcritical, L 2 -critical ( p = 4 / N + 2 ) and L 2 -supercritical perturbations. Secondly, we characterize perturbation limit behaviors of ground states u ε , ρ as ε → 0 + and find that the ε -blow-up phenomenon happens for ρ ≥ ρ c = ‖ Q ‖ L 2 2 , where Q is a positive radially symmetric ground state of - Δ u + u - (| x | - 2 ∗ | u | 2) u = 0 in R N . We prove that ∫ R N | ∇ u ε , ρ (x) | 2 d x ∼ ε - 4 N (p - 2) + 4 for ρ = ρ c and 2 < p < 2 ∗ , while ∫ R N | ∇ u ε , ρ | 2 d x ∼ ε - 4 N (p - 2) - 4 for ρ > ρ c and 4 / N + 2 < p < 2 ∗ , and obtain two different blow-up profiles corresponding to two limit equations. Finally, we study the limit behaviors as ε → + ∞ , which corresponds to a Thomas–Fermi limit. The limit profile is given by the Thomas–Fermi minimizer u TF = μ TF - | x | 2 + 1 p - 2 , where μ TF is a suitable Lagrange multiplier with exact value. Moreover, we obtain a sharp vanishing rate for u ε , ρ that ‖ u ε , ρ ‖ L ∞ ∼ ε - N N (p - 2) + 4 as ε → + ∞ . [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of [11] in the setting of the standard cubic-quintic NLS. [ABSTRACT FROM AUTHOR]

We study the existence of ground states (minimizers of an energy under a fixed masses constraint) for a system of coupled NLS equations with double power nonlinearities (classical and point). We prove that the presence of at least one subcritical power parameter guarantees existence of a ground state for masses below specific values. Moreover, we show that each ground state is given by a pair of strictly positive functions (up to rotation). Using the concentration-compactness approach, under certain restrictions we prove the compactness of each minimizing sequence. One of the principal peculiarities of the model is that in presence of critical power parameters existence of ground states depends on concrete mass parameters related with the optimal function for the Gagliardo–Nirenberg inequality. [ABSTRACT FROM AUTHOR]

In this paper, we look for normalized solutions to the following non-autonomous Schrödinger equation - Δ u = λ u + h (x) | u | q - 2 u + | u | 2 ∗ - 2 u in R N , ∫ R N | u | 2 d x = a , where N ≥ 3 , a > 0 , λ ∈ R , h ≠ c o n s t and 2 ∗ = 2 N N - 2 is the Sobolev critical exponent. In the L 2 -subcritical regime (i.e. 2 < q < 2 + 4 N ), by proposing some new conditions on h, we verify that the corresponding Pohozaev manifold is a natural constraint and establish the existence of normalized ground states. Compared to the L 2 -subcritical regime, it is necessary to apply some reverse conditions to h provided that at least L 2 -critical regime (i.e. 2 + 4 N ≤ q < 2 ∗ ) is considered. We prove the existence of minimizer on the Pohozaev manifold of the associated energy functional and determine that the minimizer is a normalized solution by using the classical deformation lemma. In particular, by imposing further assumptions on h, the ground states can be obtained. [ABSTRACT FROM AUTHOR]

The aim of this work is to present new spectral tools for studying the orbital stability of standing waves solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on looping edge graphs, namely, a graph consisting of a circle with several half-lines attached at a single vertex. The main novelty of this paper is at least twofold: by considering δ -type boundary conditions at the vertex, the extension theory of Krein &von Neumann, and a splitting eigenvalue method, we identify the Morse index and the nullity index of a specific linearized operator around of a priori positive single-lobe state profile for every positive power, this information will be main for a local stability study; and so via a bifurcation analysis on the phase plane we build at least two families of positive single-lobe states and we study the stability properties of these in the subcritical, critical, and supercritical cases. Our results recover some spectral studies in the literature associated to the NLS on looping edge graphs which were obtained via variational techniques. [ABSTRACT FROM AUTHOR]

This study presents a new approach for accurately obtaining solitary wave solutions in the Boiti–Leon–Pempinelli (BLP) system, which was developed by Italian mathematicians Luca Boiti, Franco Leon, and Guido Pempinelli in the late 1980 s. The BLP system is very significant in physics and is employed in several fields. It is used to simulate the behavior of nonlinear media in electromagnetic and acoustic wave propagation, as well as to understand the features of Bose–Einstein condensates. This research thoroughly examines the acquired solutions, with a particular focus on crucial factors such as the maximum amplitude, duration, and velocity of the waves. The accurate solitary wave solutions play a crucial role in enhancing the comprehension of these physical phenomena, establishing the foundation for possible practical uses. The suggested modified exponential function approach outperforms previous numerical strategies in terms of efficacy and dependability, demonstrating the significant potential for future study in this field. The results of our investigation are graphically shown using meticulously designed graphs, providing valuable insights into the behavior of various solutions inside the (x, t) domain. This visual depiction facilitates the identification of key characteristics such as velocity, magnitude, and form, hence improving the clarity of our study findings. [ABSTRACT FROM AUTHOR]

This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering. [ABSTRACT FROM AUTHOR]

This paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation i u ˙ - (- Δ) γ u = ± | x | ρ | u | p - 1 u , where 0 < γ < 1 and ρ < 0 . Here, one considers the inter-critical regime 0 < s c : = N 2 - 2 γ + ρ p - 1 < γ , where s c is the energy critical exponent, which is the only one real number satisfying ‖ κ 2 γ + ρ p - 1 u 0 (κ ·) ‖ H ˙ s c = ‖ u 0 ‖ H ˙ s c . In order to avoid a loss of regularity in Strichartz estimates, one assumes that the datum is spherically symmetric. First, using a sharp Gagliardo–Nirenberg-type estimate, one develops a local theory in the space H ˙ γ ∩ H ˙ s c . Then, one investigates the L N (p - 1) ρ + 2 γ concentration of finite-time blow-up solutions bounded in H ˙ s c . Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space H ˙ s c , the main difficulty here is to avoid the mass conservation law. [ABSTRACT FROM AUTHOR]

This paper studies the inhomogeneous defocusing coupled Schrödinger system i ⁢ u ˙ j + Δ ⁢ u j = | x | - ρ ⁢ (∑ 1 ≤ k ≤ m a j ⁢ k ⁢ | u k | p ) ⁢ | u j | p - 2 ⁢ u j , ρ > 0 , j ∈ [ 1 , m ] . The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ⁢ (ℝ N) such that x ⁢ f ∈ L 2 ⁢ (ℝ N) . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x ⁢ u 0 ∈ L 2 enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 , where p 0 is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ⁢ ∥ u ⁢ (t) ∥ L r ⁢ (ℝ N) ≲ 1 for some α > 0 and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ ⁢ u 0 . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 , which restricts the space dimensions to N ≤ 3 . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 . [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity. [ABSTRACT FROM AUTHOR]

In a recent paper Li et al. (Z Angew Math Phys 73:52, 2022. https://doi.org/10.1007/s00033-022-01681-4) have considered a generalized nonlinear Schrödinger equation which has extensive applications in various fields of physics and engineering. After proving Liouville integrability of this equation, they investigated the phenomenon of the modulational instability for the possible reason of the formation of the rogue waves. By means of the generalized ( 2 , N - 2 )-fold Darboux transformation, authors presented several mixed localized wave solutions, such as breathers, rogue waves and semi-rational solitons for their model equation, and accurately analyzed a number of important physical quantities. It is the aim of this Comment to point out that (i) the baseband modulation instability was developed in a wrong way and (ii) one of the two different types of Taylor series expansions for solution of Lax pair used in that article for building analytical solutions, especially the one obtained with ξ j = Z does not correspond to any solution of the spectral problem (2.1) when using u 0 x , t as the seed solution. Consequently, all mixed localized solutions that involve the mentioned Taylor series are invalid. [ABSTRACT FROM AUTHOR]

This work is dedicated to the study of existence of solutions for a Schrödinger–Poisson type system involving possible different integrodifferential operators in the presence of a nonnegative but not bounded away from zero potential. We consider a pure power nonlinearity u ↦ | u | p - 2 u , with 4 - β < p < 6 / (3 - 2 α) , where α and β are positive parameters described in each equation. Our argument is variational in nature and involves the use of a deformation type lemma on a minimization over a Nehari–Pohozaev manifold. Some regularity results are provided and a maximum type principle for integrodifferential operators is proved. [ABSTRACT FROM AUTHOR]

The study of nonlinear Schrödinger-type equations with nonzero boundary conditions introduces challenging problems both for the continuous (partial differential equation) and the discrete (lattice) counterparts. They are associated with fascinating dynamics emerging by the ubiquitous phenomenon of modulation instability. In this work, we consider the discrete nonlinear Schrödinger equation with linear gain and nonlinear loss. For the infinite lattice supplemented with nonzero boundary conditions, which describe solutions decaying on the top of a finite background, we provide a rigorous proof that for the corresponding initial boundary value problem, solutions exist for any initial condition, if and only if, the amplitude of the background has a precise value A ∗ defined by the gain-loss parameters. We argue that this essential property of this infinite lattice cannot be captured by finite lattice approximations of the problem. Commonly, such approximations are provided by lattices with periodic boundary conditions or as it is shown herein, by a modified problem closed with Dirichlet boundary conditions. For the finite-dimensional dynamical system defined by the periodic lattice, the dynamics for all initial conditions are captured by a global attractor. Analytical arguments corroborated by numerical simulations show that the global attractor is trivial, defined by a plane wave of amplitude A ∗ . Thus, any instability effects or localized phenomena simulated by the finite system can be only transient prior the convergence to this trivial attractor. Aiming to simulate the dynamics of the infinite lattice as accurately as possible, we study the dynamics of localized initial conditions on the constant background and investigate the potential impact of the global asymptotic stability of the background with amplitude A ∗ in the long-time evolution of the system. [ABSTRACT FROM AUTHOR]

Concentric water waves are ubiquitous in nature and, under certain conditions, may exhibit azimuthal instabilities. Whereas transverse instability of their plane counterparts, governed by nearly plane Korteweg–de Vries and nonlinear Schrödinger equations in the shallow and deep water limits, respectively, have been extensively studied as enabled by the existence of localized solitons, in the cylindrical case stability analysis is impeded by the corresponding finite-amplitude waves being non-localized. The equations, governing such waves, emerge from a balance between dispersive and nonlinear effects and, in particular, admit non-localized solutions of self-similar form. It is the goal of the present work to study transverse stability of such solutions, in both the shallow and deep water limits, as a function of surface tension (a Weber We number). Whereas in the shallow water case, a nearly concentric Korteweg–de Vries equation has been previous deduced, a deep water weakly nonlinear model has not been established yet. With a systematic derivation in cylindrical coordinates, we demonstrate that the appropriate envelope equation becomes of the nearly concentric nonlinear Schrödinger type with time-dependent coefficients. Transverse stability analyses of the concentric self-similar solutions to these models indicate crucial differences from the plane configurations revealing the effects of cylindrical geometry, direction of the base-state wave propagation, and its amplitude variation. In particular, the azimuthal perturbations do not admit, in general, a regular periodic structure, except for the short-wavelength perturbations on deep water which are either stretched or compressed with time as τ 1 / 2 depending on whether the base state concentric wave is traveling outward or inward. In the shallow-water case, transient azimuthal growth of outgoing finite-amplitude concentric waves is observed for all We in contrast to the plane solitons case, in which there exists a nonzero critical We; also, inward-traveling cylindrical waves are unstable for all values of We. In the deep-water case, azimuthal perturbations of outward-traveling cylindrical wave are stable in the long-time limit, but experience transient growth for short times when W e < 1 2 . As for inward-traveling waves, transverse perturbations grow for all We. These observations are at variance with the plane deep water geometry, in which transverse instability is observed for all We. [ABSTRACT FROM AUTHOR]

In this paper, the orbital instability theorem is introduced for Hamiltonian partial differential equation (PDE) systems. Our specific focus lies on the Schrödinger system featuring quadratic nonlinearity, and we apply the theorem to analyze its behavior. Our theorem establishes the abstract instability theorem for a specific class of Hamiltonian PDE systems. We consider the energy functional to be of class C 2 rather than C 3 , particularly when the second derivative of the energy exhibits multiple degenerate kernels. Using this theorem, we provide a comprehensive classification of the stability and instability of the semitrivial solution within the Hamiltonian PDE system featuring quadratic nonlinearity. This classification resolves an open problem previously posed by Colin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:2211–2226, 2009), specifically in cases of homogeneous nonlinearity. Additionally, we present proof of instability results for synchronous solutions of Hamiltonian PDE systems. We believe that this abstract theorem constitutes a novel contribution with potential applicability in various situations beyond those specifically discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of periodic Schrödinger lattice systems with spectrum 0 and saturable nonlinearities. The existence of ground state solitons of the systems under weak assumptions is obtained. The main novelties are as follows. (1) Some new sufficient conditions for the existence of ground state solitons under the “spectral endpoint” assumption are constructed. (2) Our “non-monotonic” conditions make the proofs of the boundedness of the (PS) sequences to be easier. (3) Our result extends and improves the related results in the literature. Besides, some examples are given to illuminate our result. [ABSTRACT FROM AUTHOR]

This paper is concerned with a planar Schrödinger–Poisson system involving Stein–Weiss nonlinearity 0.1 - Δ u + V (x) u + ϕ u = 1 | x | β ∫ R 2 F (u (y)) | x - y | μ | y | β d y f (u) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , and its degenerate case 0.2 - Δ u + ϕ u = ∫ R 2 F (u (y)) | x - y | μ d y f (u) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where β ≥ 0 , 0 < μ < 2 , 2 β + μ < 2 , V ∈ C (R 2 , R) and f is of exponential critical growth. By combining variational methods, Stein–Weiss inequality and some delicate analysis, we derive the existence of ground state solution for the first system. Under some mild assumptions, we introduce the Pohozaev identity of the equivalent equation of the second system and use Jeanjean's monotonicity method to achieve the existence of nontrivial solution for the second system. [ABSTRACT FROM AUTHOR]

This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four distinct exact and explicit solutions, along with the corresponding first integrals. Explanations and presentations of solutions are given in a logical manner. We derive an analytical expression for the instability gain and examine its key features using linear stability analysis. Finally, we compare the correctness of the analytical and numerical solutions. We demonstrate the robustness and stability of solitary waves through numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with time global behavior of solutions to nonlinear Schrödinger equation with a non-vanishing condition at the spatial infinity. Under a non-vanishing condition, it would be expected that the behavior is determined by the shape of the nonlinear term around the non-vanishing state. To observe this phenomenon, we introduce a generalized version of the Gross-Pitaevskii equation, which is a typical equation involving a non-vanishing condition, by modifying the shape of nonlinearity around the non-vanishing state. It turns out that, if the nonlinearity decays fast as a solution approaches to the non-vanishing state, then the equation admits a global solution which scatters to the non-vanishing element for both time directions. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem of the nonlinear Schrödinger equation on the half plane with a nonlinear Neumann boundary condition. After establishing the boundary Strichartz estimate in L 2 (R + 2) and H s (R + 2) , we consider the time local well-posedness of the problem in L 2 (R + 2) and H s (R + 2) . [ABSTRACT FROM AUTHOR]

This work is concerned with the following Klein-Gordon-Maxwell system: −Δu+V(x)u−(2ω+ϕ)ϕu=f(u),x∈R2,Δϕ=(ω+ϕ)u2,x∈R2,\left\{\begin{array}{ll}-\Delta u+V\left(x)u-\left(2\omega +\phi )\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\end{array}\right. where ω>0\omega \gt 0 is a constant, u,ϕ:R2→Ru,\phi :{{\mathbb{R}}}^{2}\to {\mathbb{R}}, V∈C(R2,R)V\in {\mathcal{C}}\left({{\mathbb{R}}}^{2},{\mathbb{R}}), and f∈C(R,R)f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) obeys exponential critical growth in the sense of the Trudinger-Moser inequality. We give some new sufficient conditions on ff, specifically related to exponential growth, to obtain the existence of nontrivial solutions. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level.