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Juncheng Wei - One of the best experts on this subject based on the ideXlab platform.
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Non-radial solutions to a bi-harmonic equation with Negative Exponent
Calculus of Variations and Partial Differential Equations, 2019Co-Authors: Ali Hyder, Juncheng WeiAbstract:We prove the existence of smooth non-radial entire solution to $$\begin{aligned} \Delta ^2 u+u^{-q}=0\quad \text {in }\mathbb {R}^3,\quad u>0, \end{aligned}$$for $$q>1$$. This answers an open question raised by McKenna and Reichel (Electron J Differ Equ 37:1–3, 2003).
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Non-radial solutions to a bi-harmonic equation with Negative Exponent.
arXiv: Analysis of PDEs, 2019Co-Authors: Ali Hyder, Juncheng WeiAbstract:We prove the existence of non-radial entire solution to $$\Delta^2 u+u^{-q}=0\quad\text{in }\mathbb{R}^3,\quad u>0,$$ for $q>1$. This answers an open question raised by P. J. McKenna and W. Reichel (E. J. D. E. \textbf{37} (2003) 1-13).
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Qualitative analysis of rupture solutions for a MEMS problem
Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2016Co-Authors: Juan Dávila, Kelei Wang, Juncheng WeiAbstract:We prove sharp Holder continuity and an estimate of rupture sets for sequences of solutions of the following nonlinear problem with Negative Exponent
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Qualitative Analysis of Rupture Solutions for an MEMS Problem
arXiv: Analysis of PDEs, 2013Co-Authors: Juan Dávila, Kelei Wang, Juncheng WeiAbstract:We prove a sharp H\"older continuity estimates of rupture sets for sequences of solutions of the following nonlinear problem with Negative Exponent $$ \Delta u= \frac{1}{u^p}, \ p>1, \ \mbox{in} \ \Omega .$$ As a consequence, we prove the existence of rupture solutions with isolated ruptures in a bounded convex domain in $\R^2$.
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on a fourth order nonlinear elliptic equation with Negative Exponent
Siam Journal on Mathematical Analysis, 2009Co-Authors: Zongming Guo, Juncheng WeiAbstract:We consider the following nonlinear fourth order equation: $T\Delta u-D\Delta^2u=\frac{\lambda}{(L+u)^2}$, $-L 0$ is a parameter. This nonlinear equation models the deflection of charged plates in electrostatic actuators under the pinned boundary condition (Lin and Yang [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), pp. 1323–1337]). Lin and Yang proved that there exists a $\lambda_c>0$ such that for $\lambda>\lambda_c$ there is no solution, while for $\lambda<\lambda_c$ there is a branch of maximal solutions. In this paper, we show that in the physical domains (two or three dimensions) the maximal solution is unique and regular at $\lambda=\lambda_c$. In a two-dimensional (2D) convex smooth domain, we also establish the existence of a second mountain-pass solution for $\lambda\in(0,\lambda_c)$. The asymptotic behavior of the second solution is also studied. The main difficulty is the analysis of the touch-down behavior.
Zongming Guo - One of the best experts on this subject based on the ideXlab platform.
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A perturbed fourth order elliptic equation with Negative Exponent
Discrete & Continuous Dynamical Systems - B, 2018Co-Authors: Zongming Guo, Long WeiAbstract:By a new type of comparison principle for a fourth order elliptic problem in general domains, we investigate the structure of positive solutions to Navier boundary value problems of a perturbed fourth order elliptic equation with Negative Exponent, which arises in the study of the deflection of charged plates in electrostatic actuators in the modeling of electrostatic micro-electromechanical systems (MEMS). It is seen that the structure of solutions relies on the boundary values. The global branches of solutions to the Navier boundary value problems are established. We also show that the behaviors of these branches are relatively "stable" with respect to the Navier boundary values.
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Morse index and symmetry breaking for an elliptic equation with Negative Exponent in expanding annuli
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2017Co-Authors: Zongming Guo, Linfeng Mei, Zhitao ZhangAbstract:Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with Negative Exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.
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Liouville type results for a p-Laplace equation with Negative Exponent
Acta Mathematica Sinica English Series, 2016Co-Authors: Zongming Guo, Linfeng MeiAbstract:Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 0, are classified via their Morse indices. It is seen that there is a critical power q = qc such that this equation has no positive radial entire solution that has finite Morse index when q > qc but it admits a family of stable positive radial entire solutions when 0 qc. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.
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Sub-harmonicity, monotonicity formula and finite Morse index solutions of an elliptic equation with Negative Exponent
Science China Mathematics, 2015Co-Authors: Zongming Guo, Feng ZhouAbstract:A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with “Negative Exponent” is established. It is well known that such a monotonicity formula plays an essential role in the study of finite Morse index solutions of equations with “positive Exponent”. Unlike the positive Exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with Negative Exponent and obtaining sharp results for their asymptotic behaviors.
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Revisit the biharmonic equation with Negative Exponent in lower dimensions
2014Co-Authors: Zongming Guo, Baishun LaiAbstract:Let $B \subset \mathbb{R}^N$ be the unit ball. We study structure of solutions to the following semilinear biharmonic problem $\Delta^2 u =\lambda (1-u)^{-p}$ in $B$ under Dirichlet or Navier boundary conditions, where $p, \lambda>0$. This arises in the study of the deflectionof charged plates in electrostatic actuators. We study in particularthe structure of solutions for $N=2$ or $3$, and show the existence of mountain-passsolutions under suitable conditions on $p$. Our results contribute to complete the picture of solutions in previous works. Moreover, we also analyze the asymptotic behavior of the constructedmountain-pass solutions as $\lambda \to 0$.
Nicolas Yunes - One of the best experts on this subject based on the ideXlab platform.
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solar system constraints on massless scalar tensor gravity with positive coupling constant upon cosmological evolution of the scalar field
Physical Review D, 2017Co-Authors: David K Anderson, Nicolas YunesAbstract:Scalar-tensor theories of gravity modify General Relativity by introducing a scalar field that couples non-minimally to the metric tensor, while satisfying the weak-equivalence principle. These theories are interesting because they have the potential to simultaneously suppress modifications to Einstein's theory on Solar System scales, while introducing large deviations in the strong field of neutron stars. Scalar-tensor theories can be classified through the choice of conformal factor, a scalar that regulates the coupling between matter and the metric in the Einstein frame. The class defined by a Gaussian conformal factor with Negative Exponent has been studied the most because it leads to spontaneous scalarization (i.e. the sudden activation of the scalar field in neutron stars), which consequently leads to large deviations from General Relativity in the strong field. This class, however, has recently been shown to be in conflict with Solar System observations when accounting for the cosmological evolution of the scalar field. We study whether this remains the case when the Exponent of the conformal factor is positive, as well as in another class of theories defined by a hyperbolic conformal factor. We find that in both of these scalar-tensor theories, Solar System tests are passed only in a very small subset of parameter space, for a large set of initial conditions compatible with Big Bang Nucleosynthesis. However, while we find that it is possible for neutron stars to scalarize, one must carefully select the coupling parameter to do so, and even then, the scalar charge is typically two orders of magnitude smaller than in the Negative Exponent case. Our study suggests that future work on scalar-tensor gravity, for example in the context of tests of General Relativity with gravitational waves from neutron star binaries, should be carried out within the positive coupling parameter class.
Juan Dávila - One of the best experts on this subject based on the ideXlab platform.
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Qualitative analysis of rupture solutions for a MEMS problem
Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2016Co-Authors: Juan Dávila, Kelei Wang, Juncheng WeiAbstract:We prove sharp Holder continuity and an estimate of rupture sets for sequences of solutions of the following nonlinear problem with Negative Exponent
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Qualitative Analysis of Rupture Solutions for an MEMS Problem
arXiv: Analysis of PDEs, 2013Co-Authors: Juan Dávila, Kelei Wang, Juncheng WeiAbstract:We prove a sharp H\"older continuity estimates of rupture sets for sequences of solutions of the following nonlinear problem with Negative Exponent $$ \Delta u= \frac{1}{u^p}, \ p>1, \ \mbox{in} \ \Omega .$$ As a consequence, we prove the existence of rupture solutions with isolated ruptures in a bounded convex domain in $\R^2$.
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On finite Morse index solutions of two equations with Negative Exponent
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2013Co-Authors: Juan DávilaAbstract:We consider the following equations involving Negative Exponent: where p > 0. Under optimal conditions on the parameters α > −2 and p > 0, we prove the non-existence of finite Morse index solution on exterior domains or near the origin. We also prove an optimal regularity result for solutions with finite Morse index and isolated rupture at 0.
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On finite Morse index solutions of two equations with Negative Exponent
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2013Co-Authors: Juan DávilaAbstract:In this note, we consider the following equations involving Negative Exponent:$$\Delta u = |x|^\alpha u^{-p},\quad u> 0\quad \text{in } \Omega \subset \mathbb{R}^n$$and$$\Delta u = u^{-p} - 1,\quad u> 0\quad \text{in } \Omega \subset \mathbb{R}^n$$where $p > 0$. Under optimal conditions on the parameters $\alpha > -2$ and $p>0$, we prove the nonexistence of finite Morse index solution on exterior domains or near the origin. We prove also an optimal regularity result for solutions with finite Morse index and isolated rupture at $0$.
Chuangbing Zhou - One of the best experts on this subject based on the ideXlab platform.
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an empirical failure criterion for intact rocks
Rock Mechanics and Rock Engineering, 2014Co-Authors: Jun Peng, Guan Rong, Ming Cai, Xiaojiang Wang, Chuangbing ZhouAbstract:The parameter m i is an important rock property parameter required for use of the Hoek–Brown failure criterion. The conventional method for determining m i is to fit a series of triaxial compression test data. In the absence of laboratory test data, guideline charts have been provided by Hoek to estimate the m i value. In the conventional Hoek–Brown failure criterion, the m i value is a constant for a given rock. It is observed that using a constant m i may not fit the triaxial compression test data well for some rocks. In this paper, a Negative Exponent empirical model is proposed to express m i as a function of confinement, and this exercise leads us to a new empirical failure criterion for intact rocks. Triaxial compression test data of various rocks are used to fit parameters of this model. It is seen that the new empirical failure criterion fits the test data better than the conventional Hoek–Brown failure criterion for intact rocks. The conventional Hoek–Brown criterion fits the test data well in the high-confinement region but fails to match data well in the low-confinement and tension regions. In particular, it overestimates the uniaxial compressive strength (UCS) and the uniaxial tensile strength of rocks. On the other hand, curves fitted by the proposed empirical failure criterion match test data very well, and the estimated UCS and tensile strength agree well with test data.
