In this article, we consider a singular elliptic problem with singular nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent. By using variational methods and Palais-Smale condition, we show ...the existence of at least two nontrivial solutions. The result depends crucially on the parameters \(a,b,N,\beta,\gamma,\lambda,\mu\).
For more information see https://ejde.math.txstate.edu/Volumes/2023/54/abstr.html
Abstract In this paper, a class of quasilinear Schrödinger equations with discontinuous nonlinearity is considered. After changing variables, by using nonsmooth critical point theory, we obtain the ...existence and concentration of positive solutions for this problem under suitable conditions. Our results cover and extend some results for these differentiable quasilinear Schrödinger problems.
A combination of variational and empirical methods to determine the porosity is used to give a theoretical explanation of the power-law dependence of the dimensionless velocity of the constrained ...motion of particles during sedimentation.
This paper aims to research the effects of overlapping deformation and the reduction in shear stiffness of corrugated steel webs, caused by the accordion effect, on the shear stress and shear ...capacity ratio of variable cross-section corrugated steel webs. Firstly, three angular displacement functions were introduced to account for overlapping deformation and obtain the shear strain of the corrugated steel web. The resulting shear stress was then calculated by considering the reduction in shear stiffness. Secondly, the combination of the finite beam segment method and energy variational method was utilized to derive the stiffness matrix and nodal load array for analyzing composite box beam. By solving for the angular displacement function, both shear stress and shear capacity ratio can be obtained. Finally, various factors such as corrugated steel web types, load forms, structural types, and boundary conditions were examined through examples to analyze their effects on both shear stress and shear ratio. The results show that, the overlapping deformation weakens the shear strain to a certain extent. When considering the reduction in shear stiffness before and after, the deviation in shear stress results for different types of corrugated steel webs ranged from 7.5% to 13.2%. Moreover, a decrease in calculation error was noted with a rise in single-wavelength. For a statically determinate cantilever beam, regardless of load conditions, the direction of maximum and minimum shear stresses remains consistent within the web, with corresponding maximum and minimum shear capacity ratios of 77.37% and 40.96%, respectively. In contrast, a single-span statically indeterminate beam exhibits alternating directions of shear stress, with corresponding maximum and minimum shear capacity ratios being 197.63% and 32.32%, respectively.
A novel variational meshless method is presented to calculate a large number of modes of a homogeneous waveguide with an arbitrary shape. The proposed technique is based on the variational principle ...combined with the meshless method using radial basis functions (RBFs). Compared with the more traditional point-matching meshless method with RBFs, the proposed technique leads to a well-conditioned matrix problem, and no preconditioning is needed. An iterative refinement algorithm is also presented, which generates automatically new collocation points and locally defined shape parameters for the RBFs, until a prescribed convergence is reached. As proved through many examples, including waveguides with sharp corners, the variational meshless method permits to calculate a large number of waveguide modes with high precision in a very short computing time.
In this paper, we consider a class of nonlinear nonlocal problem involving an anisortropic operator and an external potential. We show the existence of positive and sign-changing solutions of this ...problem via the variational methods.
This paper considers an optimal boundary control problem for fluid pipelines with terminal valve control. The goal is to minimize pressure fluctuation during valve closure, thus mitigating water ...hammer effects. We model the fluid flow by two coupled hyperbolic PDEs with given initial conditions and a boundary control governing valve actuation. To solve the optimal boundary control problem, we apply the control parameterization method to approximate the time-varying boundary control by a linear combination of basis functions, each of which depends on a set of decision parameters. Then, by using variational principles, we derive formulas for the gradient of the objective function (which measures pressure fluctuation) with respect to the decision parameters. Based on the gradient formulas obtained, we propose a gradient-based optimization method for solving the optimal boundary control problem. Numerical results demonstrate the capability of optimal boundary control to significantly reduce pressure fluctuation.
In this study, the variational method concerning displacement components is applied to solve the large deformation problem of a thin cylindrical shell with its four sides fully fixed and under ...uniformly distributed loads, in which the material that constitutes the shell has a bimodular effect, in comparison to traditional materials, that is, the material will present different moduli of elasticity when it is in tension and compression. For the purpose of the use of the displacement variational method, the physical equations on the bimodular material model and the geometrical equation under large deformation are derived first. Thereafter, the total strain potential energy is expressed in terms of the displacement component, thus bringing the possibilities for the classical Ritz method. Finally, the relationship between load and central deflection is obtained, which is validated with the numerical simulation, and the jumping phenomenon of thin cylindrical shell with a bimodular effect is analyzed. The results indicate that the bimodular effect will change the stiffness of the shell, thus resulting in the corresponding change in the deformation magnitude. When the shell is relatively thin, the bimodular effect will influence the occurrence of the jumping phenomenon of the cylindrical shell.
In this paper, we propose a modified variational approach to predict the morphology of the flexible nozzle used in wind tunnel. Different from previous studies, the movements of the multiple hinges ...are considered as movable displacement boundary conditions during establishing the potential energy functional. The cubic spline interpolation method is employed to supply the supplementary boundary conditions in calculation of the functional minimization problem. Current analytical model is verified by experiments carried out on a fixed-flexible nozzle structure whose geometries and materials are the same as those from a commissioned supersonic nozzle. The maximum deviation between the predictions from theoretical method and laser displacement testing does not exceed 0.5 mm. This method can also deal with the large deflection beam problem with multiple movable boundaries.
PDF
