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In this article we consider the following fractional semilinear elliptic equation (-Δ)s u + |x|² u = ωu + |u|²σ in RN, where s ∈ (0, 1), N > 2s, σ ∈ (0, 2s/N-2s) and ω ∈ (0, λ1). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following planar Schrödinger–Poisson system 0.1 - Δ u + a (x) u + 2 π ϕ u = | u | p - 2 u e α 0 | u | γ , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where p > 2 , α 0 > 0 and 0 < γ ≤ 2 , the potential a : R 2 → R is invariant under the action of a closed subgroup of the orthogonal transformation group O(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if 0 < γ < 2 and p > 2 . Furthermore, in the critical case γ = 2 and p > 4 , we prove that equation (0.1) possesses a positive solution which is invariant under the same group action. [ABSTRACT FROM AUTHOR]

This article deals with the following fractional N s -Laplace Kichhoff equation involving exponential growth of the form: ε N K [ u ] s , N s N s (- Δ) N / s s u + Z (x) | u | N s - 2 u = f (u) in R N , where ε > 0 is a parameter, s ∈ (0 , 1) and (- Δ) p s is the fractional p-Laplace operator with p = N s ≥ 2 , K is a Kirchhoff function, f is a continuous function with exponential growth and Z is a potential function possessing a local minimum. Under some suitable conditions, we obtain the existence, multiplicity and concentration of solutions to the above problem via penalization methods and Lyusternik-Schnirelmann theory. [ABSTRACT FROM AUTHOR]

This paper investigates a class of planar Schrödinger–Poisson systems with critical exponential growth. The conditions for the local well‐posedness of the Cauchy problem in the energy space are defined. By introducing innovative ideas and relaxing some of the classical growth assumptions on nonlinearity, this study shows that such a system has at least two standing waves with a prescribed mass. One wave is a ground‐state standing wave with positive energy, and another one is a high‐energy standing wave with positive energy. In addition, with the help of the local well‐posedness, it is shown that the set of ground‐state standing waves is orbitally stable. [ABSTRACT FROM AUTHOR]

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

This study examines a class of time-dependent constitutive equations used to describe viscoelastic materials under creep in solid mechanics. In nonlinear elasticity, the strain response to the applied stress is expressed via an implicit graph allowing multi-valued functions. For coercive and maximal monotone graphs, the existence of a solution to the quasi-static viscoelastic problem is proven by applying the Browder–Minty fixed point theorem. Moreover, for quasi-linear viscoelastic problems, the solution is constructed as a semi-analytic formula. The inverse viscoelastic problem is represented by identification of a design variable from non-smooth measurements. A non-empty set of optimal variables is obtained based on the compactness argument by applying Tikhonov regularization in the space of bounded measures and deformations. Furthermore, an illustrative example is given for the inverse problem of isotropic kernel identification. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

This article is about the following Kirchhoff–Schrödinger–Poisson system with steep potential well 1.1 −(a+b∫ℝ3|∇u|2dx)△u+λV(x)u+μϕ(x)u=f(x,u)+h(x)|u|αinℝ3,−△ϕ=u2,inℝ3,$$ \left\{\begin{array}{ll}-\left(a&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;3}{\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;2 dx\right)\triangle u&#x0002B;\lambda V(x)u&#x0002B;\mu \phi (x)u&#x0003D;f\left(x,u\right)&#x0002B;h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{\alpha }& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\\ {}-\triangle \phi &#x0003D;{u}&#x0005E;2,& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\end{array}\right. $$ where a,b,λ>0$$ a,b,\lambda >0 $$ are constants, μ>0$$ \mu >0 $$, and 0<α<1,f∈C(ℝN×ℝ,ℝ)$$ 0<\alpha <1,f\in C\left({\mathrm{\mathbb{R}}}&#x0005E;N\times \mathrm{\mathbb{R}},\mathrm{\mathbb{R}}\right) $$. By using the variational principle, we overcome the difficulties caused by Poisson's term and obtain system (1.1) that has two nontrivial solutions under certain assumptions. Moreover, we study the concentration of solutions and obtain new conclusions of system (1.1). Finally, we present the case where the solution to system (1.1) does not exist. [ABSTRACT FROM AUTHOR]

A fully quantum, numerically accurate methodology is presented for the simulation of the exciton dynamics and time-resolved fluorescence of cavity-tuned two-dimensional (2D) materials at finite temperatures. This approach was specifically applied to a monolayer WSe2 system. Our methodology enabled us to identify the dynamical and spectroscopic signatures of polaronic and polaritonic effects and to elucidate their characteristic timescales across a range of exciton–cavity couplings. The approach employed can be extended to simulation of various cavity-tuned 2D materials, specifically for exploring finite temperature nonlinear spectroscopic signals. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of standing wave solutions for a class of nonlinear fractional Schrödinger-Poisson system involving nonlinearity with subcritical and critical growth. We suppose that the potential V satisfies either Palais-Smale type condition or there exists a bounded domain Ω such that V has no critical point in ∂ Ω . To overcome the "lack of compactness" of the problem, we combine Del Pino-Felmer's penalization technique with Moser's iteration method and some ideas from Alves [1]. [ABSTRACT FROM AUTHOR]

This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group : Here, , , , , and . By means of variational techniques, we extended the arguments developed in [1]. In addition, we also established the existence result for the subelliptic system which involved sub-Laplacian and critical homogeneous terms. [ABSTRACT FROM AUTHOR]

Consider time-harmonic electromagnetic wave scattering by an infinitely long, cylindrical, orthotropic dielectric partially coated with a very thin layer of a highly conductive material, which can be modeled by a transmission problem with mixed boundary conditions. Having established the well-posedness of the direct and interior transmission problem by the variational method under certain conditions, we make use of the classical linear sampling method to reconstruct the shape of the obstacle. Then, based on a modification of the general data-to-pattern operator G, we propose a novel and simple method to justify the modified linear sampling method. [ABSTRACT FROM AUTHOR]

In this paper, we study multi-bump solutions of the following Kirchhoff type equation: - M ∫ R 2 | ∇ u | 2 d x Δ u + μ V (x) + h (x) u = λ f (u) in R 2 , where M is continuous with inf R + M > 0 , V ≥ 0 and its zero set has several disjoint bounded components, μ , λ are positive parameters, f has exponential critical growth. When V decays to zero at infinity, we use variational methods to obtain the existence and concentration behavior of multi-bump solutions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a class of planar Schrödinger–Poisson systems, namely, - Δ u + V (x) u + γ 2 π (log (| · |) ∗ | u | 2) u = b | u | p - 2 u in R 2 , where γ > 0 , b ≥ 0 , p > 2 and V ∈ C (R 2 , R) is an unbounded potential function with inf R 2 V > 0 . Suppose moreover that the potential V satisfies { x ∈ R 2 : V (x) ≤ M } < ∞ for every M > 0 , we establish the existence of ground state solutions for this system via variational methods. Furthermore, we also explore the minimax characterization of ground state solutions. Our main results can be viewed as a counterpart of the results from Molle and Sardilli (Proc Edinb Soc 65:1133–1146, 2022), where the authors studied the existence of ground state solutions for the above planar Schrödinger–Poisson system in the case where b > 0 and p > 4 . [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff-type equation:... where M ∈ C(R+,R+) is a general function, V ≥ 0 and its zero set may have several disjoint connected components, μ > 0 is a parameter, K is permitted to be unbounded above, and f has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhofftype equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case M ≡ 1. [ABSTRACT FROM AUTHOR]

This study focused on establishing the existence and multiplicity of solutions for gauged nonlinear Schrödinger equations set on the plane with sign-changing potentials. Our findings contribute to the extension of recent advancements in this area of research. Initially, we examined scenarios where the potential function V is lower-bounded and the function space has a compact embedding into Lebesgue spaces. Subsequently, we addressed more complex cases characterized by a sign-changing potential V and a function space that fails to compactly embed into Lebesgue spaces. The proofs of our results are based on the Trudinger-Moser inequality, the application of variational methods, and the utilization of Morse theory. [ABSTRACT FROM AUTHOR]

In the present paper, by the variational method coupled with mountain pass type theorems, we study nontrivial homoclinic solutions of nonlinear $ (p, q) $-Laplacian partial difference equations with a parameter $ \lambda>0 $:$ \begin{equation*} \begin{split} &\Delta^2_1(\phi_{p_2}(\Delta^2_1u(k-2, l)))+\Delta^2_2(\phi_{p_2}(\Delta^2_2u(k, l-2)))-a[\Delta_1(\phi_{p_1}(\Delta_1u(k-1, l)))\\ &+\Delta_2(\phi_{p_1}(\Delta_2u(k, l-1)))]+V(k, l)\phi_q(u(k, l))\\& = \lambda f((k, l), u(k, l)). \end{split} \end{equation*} $We prove that the above equation admits two nontrivial homoclinic solutions if $ \lambda $ is sufficiently large and one nontrivial homoclinic solutions if $ \lambda>0 $. Our obtained results generalize and improve some existing ones. [ABSTRACT FROM AUTHOR]

The variational method is used to study the hard confinement of a two-particle quantum system in two potential models, the Cornell potential and the global potential, with Dirichlet-type boundary conditions at various cut-off radii. The trial wave function is constructed as the product of the 1 S free hydrogen atom wave function or 1 S harmonic oscillator wave function times a linear cut-off function of the form (r − z) to ensure hard entrapment within a sphere of radius z. The behavior of | ψ | 2 , the wave function at the origin (WFO) and the mean radius 〈 r 〉 are computed for different situations and compared for the two potential models. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of upper critical Kirchhoff‐type fractional Laplacian system with Choquard nonlinearities and magnetic fields. With the help of the limit index theory and the concentration–compactness principles for fractional Sobolev spaces, we establish the existence of infinitely many nontrivial radial solutions for the above system. A distinguished feature of this paper is that the above Kirchhoff‐type system is degenerate, that is, the Kirchhoff term is zero at zero. [ABSTRACT FROM AUTHOR]

This study aimed to investigate the elastic deformation of rotational wood-dowel welding joints by using the variational method, as there was no efficient tool to access the deformation behavior of welding joints perpendicular to axis orientation so far. Based on the variational method, the displacement control equation of axially-loaded dowel-welded wood joints was derived. By incorporating the boundary conditions and using an iterative approach, the parameters required in the control equation were determined. The analytical expression ϕ(r) for the deformation function of the wood substrate under elastic conditions was proposed, thereby determining the displacement and axial force of the single wood-dowel. The elasticity modulus was the primary influence parameter on the function ϕ(r). However, the welding depth had a more critical impact, and reasonable combination of parameters could lead to a better interaction between the wood dowel and substrate. The comparison between calculated results and test trials collected from the previous studies indicated that the elastic solution method could accurately estimate the pull-out ultimate bearing capacity and deformation characteristics of the welding joints. This work can provide unique perspective to understand the technology of rotational wood-dowel welding joints, and a good reference suggestion for the manufacture of multi-dowel welding joints. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the paper there are established many results for a transmission problem with critical Hardy-Sobolev exponential source term u 3 | x | in R 3 . We obtain that there are at least three weakly nontrivial solutions when a positive coefficient of nonhomogeneous term is enough small using the variational method. Next infinitely many classical solutions are obtained when the coefficient equals to zero. Moreover, a new compactness condition is derived with the help of Brezis-Lieb’s lemma and Mazur’s lemma. [ABSTRACT FROM AUTHOR]

This article is devoted to study a class of new p(x)p\left(x)-Kirchhoff equation. By means of perturbation technique, variational method, and the method invariant sets for the descending flow, the existence and multiplicity of solutions to this problem are obtained.

The problem deals with the anisotropic $p(x)$-Laplacian operator where $p_i$ are Lipschitz continuous functions $2\leq p_i(x)

The methodology and results of calculating the stress-strain state of metal frame structures when they are strengthened by attaching additional elements to the original ones. With such strengthening, additional assembly stresses emerge in the structure. This paper presents a mathematical model and a variational method for determining assembly displacements and stresses, where the equations for displacement of the system due to unit concentrated forces are not used in solving the problem. The proposed mathematical model and method can be used with equal success for solving linear and nonlinear problems. The mathematical model and the calculation method for analyzing the stress-strain state of a frame structure strengthened during operation allow to successively determine displacements and stresses in the structure from the effects of initial, assembly and additional operational loads. The basic hypotheses of the bar theory, taking shearing into account, and the Lagrange variational principle are applied. A distinguished feature of the calculation method is that in the process of solving the problem, constraints are imposed on the displacements of the original and strengthening structural elements and, taking into account these constraints, the assembly displacements and stresses due to the initial loads are calculated. This feature significantly simplifies the solution of the problem and allows to expand the range of questions under study, since it removes the limitations associated with the determination of assembly forces. The test problems have been solved. Comparison of the values of assembly displacements and stresses obtained in the test problems and determined by other methods demonstrate reliability and high accuracy of the calculations.

This study focused on establishing the existence and multiplicity of solutions for gauged nonlinear Schrödinger equations set on the plane with sign-changing potentials. Our findings contribute to the extension of recent advancements in this area of research. Initially, we examined scenarios where the potential function $ V $ is lower-bounded and the function space has a compact embedding into Lebesgue spaces. Subsequently, we addressed more complex cases characterized by a sign-changing potential $ V $ and a function space that fails to compactly embed into Lebesgue spaces. The proofs of our results are based on the Trudinger-Moser inequality, the application of variational methods, and the utilization of Morse theory.

In this paper, we study the following Kirchhoff-type equation:$ \begin{equation*} M\left(\displaystyle{\int}_{\mathbb{R}^2}(|\nabla u|^2 +u^2)\mathrm{d} x\right)(-\Delta u+u) + \mu V(x)u = K(x) f(u) \ \ \mathrm{in} \ \ \mathbb{R}^2, \end{equation*} $where $ M \in C(\mathbb{R}^+, \mathbb{R}^+) $ is a general function, $ V \geq 0 $ and its zero set may have several disjoint connected components, $ \mu > 0 $ is a parameter, $ K $ is permitted to be unbounded above, and $ f $ has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case $ M \equiv 1 $.

In this article, we are concerned with the nonlinear Schrödinger equation −Δu+λu=μ∣u∣p−2u+f(u),inR2,-\Delta u+\lambda u=\mu {| u| }^{p-2}u+f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{2}, having prescribed mass ∫R2∣u∣2dx=a2>0,\mathop{\int }\limits_{{{\mathbb{R}}}^{2}}{| u| }^{2}{\rm{d}}x={a}^{2}\gt 0, where λ\lambda arises as a Lagrange multiplier, μ>0\mu \gt 0, p∈(2,4]p\in \left(2,4], and the nonlinearity f∈C1(R,R)f\in {C}^{1}\left({\mathbb{R}},{\mathbb{R}}) behaves like e4πu2{e}^{4\pi {u}^{2}} as ∣u∣→+∞| u| \to +\infty . For a L2{L}^{2}-critical or L2{L}^{2}-subcritical perturbation μ∣u∣p−2u\mu {| u| }^{p-2}u, we investigate the existence of normalized solutions to the aforementioned problem. Moreover, the limiting profiles of solutions have been considered as μ→0\mu \to 0 or a→0a\to 0. This result can be considered as a supplement to the work of Soave (Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal. 279 (2020), no. 6, 1–43) and Alves et al. (Normalized solutions for a Schrödinger equation with critical growth in RN{{\mathbb{R}}}^{N}, Calc. Var. Partial Differential Equations 61 (2022), no. 1, 1–24).

In this article we consider the following fractional semilinear elliptic equation \[(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,\] where \(s\in (0,1)\), \(N\gt 2s\), \(\sigma\in (0,\frac{2s}{N-2s})\) and \(\omega\in (0, \lambda_1)\). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result.

In this article, we investigate the Rayleigh-Taylor instability in a system of two-dimensional nonhomogeneous incompressible fluid equations with Coriolis force and partial viscosity. First, we employ variational methods to construct linear unstable solutions to the corresponding linearized equations of the system. Second, we utilize the classical Osgood lemma to derive nonlinear energy estimates for the perturbed equations. The local existence of solutions to the perturbed equations is established by using the semi-Galerkin method and the expanding domain method. Finally, we prove the nonlinear instability by combining the properties of the linear unstable solutions and the nonlinear energy estimates. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger equation: - Δ u = σ f (u) + λ u , in R N , ∫ R N | u | 2 d x = a , u ∈ H 1 (R N) , where N ≥ 3 , a > 0 , σ > 0 , and λ ∈ R appears as a Lagrange multiplier. Assume that the nonlinear term f satisfies conditions only in a neighborhood of zero. For f has a subcritical growth, we prove the existence of the positive normalized solution for the equation with sufficiently small σ > 0 . For f has a supercritical growth, we derive the existence of the positive normalized solution for the equation with σ > 0 large enough. In addition, we also obtain infinitely many normalized solutions with sufficiently small σ > 0 for the subcritical case. [ABSTRACT FROM AUTHOR]

Underwater images suffer from blurred details and color distortion due to light attenuation from scattering and absorption. Current underwater image enhancement (UIE) methods overlook the effects of forward scattering, leading to difficulties in addressing low contrast and blurriness. To address the challenges caused by forward and backward scattering, we propose a novel variational-based adaptive method for removing scattering components. Our method addresses both forward and backward scattering and effectively removes interference from suspended particles, significantly enhancing image clarity and contrast for underwater applications. Specifically, our method employs a backward scattering pre-processing method to correct erroneous pixel interferences and histogram equalization to remove color bias, improving image contrast. The backward scattering noise removal method in the variational model uses horizontal and vertical gradients as constraints to remove backward scattering noise. However, it can remove a small portion of forward scattering components caused by light deviation. We develop an adaptive method using the Manhattan Distance to completely remove forward scattering. Our approach integrates prior knowledge to construct penalty terms and uses a fast solver to achieve strong decoupling of incident light and reflectance. We effectively enhance image contrast and color correction by combining variational methods with histogram equalization. Our method outperforms state-of-the-art methods on the UIEB dataset, achieving UCIQE and URanker scores of 0.636 and 2.411, respectively. [ABSTRACT FROM AUTHOR]

This article delves into the L p Minkowski problem within the framework of generalized Gaussian probability space. This type of probability space was initially introduced in information theory through the seminal works of Lutwak et al. (Ann Probab 32(1B):757–774, 2004, IEEE Trans Inf Theory 51(2):473–478, 2005), as well as by Lutwak et al. (IEEE Trans Inf Theory 58(3):1319–1327, 2012). The primary focus of this article lies in examining the existence of this problem. While the variational method is employed to explore the necessary and sufficient conditions for the existence of the normalized Minkowski problem when p ∈ R \ { 0 } , our main emphasis is on the existence of the generalized Gaussian Minkowski problem without the normalization requirement, particularly in the smooth category for p ≥ 1 . [ABSTRACT FROM AUTHOR]

We consider the following nonlinear nonlocal elliptic problem: −a+b∫R3∣∇ψ∣2dxΔψ+λψ=∫R3G(ψ(y))∣x−y∣αdyG′(ψ),x∈R3,-\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla \psi | }^{2}{\rm{d}}x\right)\Delta \psi +\lambda \psi =\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{G\left(\psi (y))}{{| x-y| }^{\alpha }}{\rm{d}}y\right)G^{\prime} \left(\psi ),\hspace{1em}x\in {{\mathbb{R}}}^{3}, where a,b>0a,b\gt 0 are constants, λ>0\lambda \gt 0 is a parameter, α∈(0,3)\alpha \in \left(0,3), and G∈C1(R,R)G\in {{\mathcal{C}}}^{1}\left({\mathbb{R}},{\mathbb{R}}). By using variational methods, we establish the existence of least energy solutions for the above equation under conditions on the nonlinearity GG we believe to be almost necessary. Some qualitative properties of the least energy solutions are also obtained

In this paper, we were concerned with the multiplicity of the large periodic solutions to a super-linear wave equation with a general variable coefficient. In general, the variable coefficient $ \rho(\cdot) $ needs to be satisfied $ \text{ess inf}\, \eta_\rho(\cdot) > 0 $ with $ \eta_\rho(\cdot) = \frac{1}{2}\frac{\rho''}{\rho}-\frac{1}{4}\big(\frac{\rho'}{\rho}\big)^2 $. Especially, the case $ \eta_\rho(\cdot) = 0 $ is presented as an open problem in [Trans. Amer. Math. 349: 2015-2048, 1997]. Here, without any restrictions on $ \eta_{\rho}(\cdot) $, we established the multiplicity of large periodic solutions for the Dirichlet-Neumann boundary condition and Dirichlet-Robin boundary condition when the period $ T = 2\pi\frac{2a-1}{b} $ with $ a, b \in \mathbb{N}^+ $. The key ingredient of the proof is the combination of the variational method and an approximation argument. Since the sign of $ \eta_\rho(\cdot) $ can change, our results can be applied to the classical wave equation.

In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system $ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $ where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.

We consider the existence of ground state solutions for a class of zero-mass Chern–Simons–Schrödinger systems − Δ u + A 0 u + ∑ j = 1 2 A j 2 u = f ( u ) − a ( x ) | u | p − 2 u , ∂ 1 A 2 − ∂ 2 A 1 = − 1 2 | u | 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , ∂ 1 A 0 = A 2 | u | 2 , ∂ 2 A 0 = − A 1 | u | 2 , where a : R 2 → R + is an external potential, p ∈ ( 1 , 2 ) and f ∈ C ( R ) denotes the nonlinearity that fulfills the critical exponential growth in the Trudinger–Moser sense at infinity. By introducing an improvement of the version of Trudinger–Moser inequality approached in (J. Differential Equations 393 (2024) 204–237), we are able to investigate the existence of positive ground state solutions for the given system using variational method. [ABSTRACT FROM AUTHOR]

We study the following gauged nonlinear Schrödinger equation { − Δ u + (∫ | x | ∞ h u (s) s u 2 (s) d s + h u 2 (| x |) | x | 2 ) u = f (u) − a | u | p − 2 u , u (x) = u (| x |) , where a > 0 , p ∈ (1 , 2) , h u (s) = ∫ 0 s r 2 u 2 (r) d r and f possesses the supercritical exponential growth in the Trudinger-Moser sense at infinity. Via introducing a new type of Trudinger-Moser inequality in a suitable work space here, we shall exploit the general minimax principle and elliptic regular result to investigate the existence of mountain-pass type solutions for the equation using variational method. [ABSTRACT FROM AUTHOR]

We conduct a comprehensive analysis of two data assimilation methods: the first utilizes the discrete adjoint approach with a correction applied to the production term of the turbulence transport equation, preserving the Boussinesq approximation. The second is a state observer method that implements a correction in the momentum equations alongside a turbulence model, both applied to fluid dynamics simulations. We investigate the impact of varying computational mesh resolutions and experimental data resolutions on the performance of these methods within the context of a periodic hill test case. Our findings reveal the distinct strengths and limitations of both methods, which successfully assimilate data to improve the accuracy of a RANS simulation. The performance of the variational model correction method is independent of input data and computational mesh resolutions. The state observer method, on the other hand, is sensitive to the resolution of the input data and CFD mesh. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in R N . With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem. [ABSTRACT FROM AUTHOR]

In this paper, by employing the variational method, we show that a boundary value problem of sixth order has infinitely many solutions. In fact, via a critical point theorem, we will present sufficient conditions such that the problem has a sequence of solutions in a suitable function space. Specific cases and an example of results are also stated. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate the effects of nonlinear geometry and a nonlinear Pasternak medium on the free vibration and the mechanical buckling of viscoelastic open-cell foam beams using a semianalytical method. The kinematic relation is considered by the Euler–Bernoulli hypothesis and von Karman strains, while constitutive relations are defined via the separable-kernel framework and Boltzmann–Volterra superposition principles. A nonlinear nonsymmetric porosity distribution through the thickness direction is simulated using a power-law relationship. The Galerkin method, variational method, and a numerical iterative algorithm are applied to solve two coupled partial differential equations with frequency-dependent coefficients. To verify our results, nonlinear frequencies and buckling loads of elastic beams, frequencies, and loss factors of viscoelastic beams are compared with available results and close correlation is observed. The influences of axial force, boundary condition, elastic medium, and a frequency-dependent constitutive relation on vibrational characteristics are scrutinized through parametric studies. [ABSTRACT FROM AUTHOR]

We investigate a class of the nonlinear Schrödinger equation in R N - Δ u + V (x) u = | u | 2 ∗ - 2 u + λ | u | p - 2 u , where N ≥ 3 , λ > 0 and p ∈ (2 , 2 ∗) with 2 ∗ = 2 N N - 2 . Here, V (x) = V 1 (x) for x 1 > 0 and V (x) = V 2 (x) for x 1 < 0 , where V 1 , V 2 are periodic in each coordinate direction. By providing a splitting Lemma corresponding to non-periodic external potential, we obtain the existence of ground state solution for the above problem. It is worth to mention that the arguments used in this paper are also valid for the Sobolev subcritical problem studied by Dohnal et al. (Commun Math Phys 308:511–542, 2011). [ABSTRACT FROM AUTHOR]

In this work, we find an interval for a parameter λ for which a functional J - λI possesses a sequence of critical points. It should be noted that members of this sequence will be weak solutions to an elliptic problem with Hardy potential. [ABSTRACT FROM AUTHOR]

The remote sensing applications demand the imaging products composed of rich spatial details and high spectral resolution, namely high resolution multispectral (HRMS) images which can not be generated by current generation satellite sensors due to technological and physical constraints. Pansharpening is an image fusion approach that combines the rich geometric details of the panchromatic image and the color information from the multispectral image into a single product. Inspite of the availability of numerous pansharpening techniques, there is a necessity to develop an effective mechanism for the balanced enhancement of spatial and spectral quality in the fused HRMS image. In this paper, a constrained optimization approach for multispectral and PAN image fusion (COAMP) is proposed. The total variation is an efficient operator to extract the geometric features of an image. The spatial differences are exported into the HRMS image with the aid of TV based gradient operator. The spectral information is transferred to the fused image using spectral vectors obtained by taking gradients in the third dimension along the bands of the multispectral image. The optimization problem is formulated and minimized using an efficient solver to get the HRMS image. The comprehensive experiments performed on three different datasets at both reduced scale and full scale resolutions. Additionally, quantitative assessment is also performed using six well-known metrics. The experiments approves the efficacy of the proposed COAMP model. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Fučik spectrum of fractional Schrödinger operators (- Δ) s + V , which is defined as the set Σ K of all (α , β) ∈ R 2 such that (- Δ) s u + V (x) u = α u + - β u - , x ∈ R N has a non-trivial solution u , where u ± = max { ± u , 0 } , N > 2 s , s ∈ (0 , 1) and V is an external unbounded potential function. Based on variational method, it turns out that the Fučik spectrum has a first non-trivial curve C being Lipschitz continuous, decreasing and with a certain asymptotic behavior. And then, as applications, we obtain the existence of non-trivial solutions for nonlinear Schrödinger equations with non-resonant nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear Schrödinger system with quadratic interaction. We extend the recent results of Fukaya et al. (Math. Ann. 2024) and show that the system has a ground state in ℝ4$$ {\mathrm{\mathbb{R}}}&#x0005E;4 $$ when the mass parameter κ$$ \kappa $$ is larger than 12$$ \frac{1}{2} $$. [ABSTRACT FROM AUTHOR]