Order Nonlinear Elliptic Equations Research Articles

Higher order mKdV breathers: nonlinear stability

We are interested in stability results for breather solutions of the 5th, 7th and 9th order mKdV equations. We show that these higher order mKdV breathers are stable in $H^2(\R)$, in the same way as \emph{classical} mKdV breathers. We also show that breather solutions of the 5th, 7th and 9th order mKdV equations satisfy the same stationary fourth order nonlinear elliptic equation as the mKdV breather, independently of the order, 5th, 7th or 9th, considered.

Regularity of Hamiltonian stationary equations in symplectic manifolds

In this paper, we prove that any C1-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond.

Solving second order non-linear elliptic partial differential equations using generalized finite difference method

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (pde’s): wave propagation, advection–diffusion, plates, beams, etc.The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear elliptic pde’s.This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the pde’s are based on the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Also it is shown that in 2D a regular neighbourhood of eight nodes can be regarded as a generalization of a classical finite difference formula with a sixth order truncation error.This paper shows the application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer.

Asymptotic bifurcation and second order elliptic equations on [formula omitted

This paper deals with asymptotic bifurcation, first in the abstract setting of an equation G(u)=λu, where G acts between real Hilbert spaces and λ∈R, and then for square-integrable solutions of a second order non-linear elliptic equation on RN. The novel feature of this work is that G is not required to be asymptotically linear in the usual sense since this condition is not appropriate for the application to the elliptic problem. Instead, G is only required to be Hadamard asymptotically linear and we give conditions ensuring that there is asymptotic bifurcation at eigenvalues of odd multiplicity of the H-asymptotic derivative which are sufficiently far from the essential spectrum. The latter restriction is justified since we also show that for some elliptic equations there is no asymptotic bifurcation at a simple eigenvalue of the H-asymptotic derivative if it is too close to the essential spectrum.

Maximum Principles for Fourth Order Nonlinear Elliptic Equations with Applications

The paper is devoted to prove maximum principles for a certain func- tionals defined for solution of the fourth order nonlinear elliptic equation. The max- imum principle so obtained is used to prove the nonexistence of nontrivial solutions of the fourth order nonlinear elliptic equation with some zero boundary conditions. Hopf's maximum principle is the main ingredient.

Liouville-type theorems for the fourth order nonlinear elliptic equation

In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equationΔ2u=|x|a|u|p−1uin Ω, where a⩾0, p>1 and Ω⊂Rn is an unbounded domain of Rn, n⩾5. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence, which is used to obtain sharp results.

Теорема типа Фрагмена — Линделефа для нелинейных эллиптических уравнений высокого порядка

Теорема типа Фрагмена — Линделефа для нелинейных эллиптических уравнений высокого порядка

The Gel’fand Problem for the Biharmonic Operator

We study stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains.

On iterative schemes for a stationary problem to a quantum drift diffusion model

In this paper, we propose a new iterative scheme to obtain a stationary solution to a quantum drift diffusion (QDD) model for semiconductors in a multi-dimensional space. The model consists of two fourth order nonlinear elliptic equations for the carrier densities and the nonlinear Poisson equation for the electrostatic potential. The present scheme adopts the Gummel method to decouple the system. For the computational efficiency, this scheme also employs an exponential transformation of variables to preserve positivity of the carrier densities and an inner iteration to couple the quantum potentials with the electrostatic potential. The present scheme is constructed to improve the other schemes developed by Ancona, de Falco and Odanaka. We show the advantage of the present scheme on computational costs in comparison to the other schemes for simulations of current-voltage characteristics for a double-gate MOSFET.

On positive solutions for a fourth order asymptotically linear elliptic equation under Navier boundary conditions

A result on existence of positive solution for a fourth order nonlinear elliptic equation under Navier boundary conditions is established. The nonlinear term involved is asymptotically linear both at the origin and at infinity. We exploit topological degree theory and global bifurcation.

Nonlinear elliptic fourth order equations existence and multiplicity results

This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold. In the case of constant coefficients we obtain the existence of three distinct solutions.

Maximum principles and bounds for a class of fourth order nonlinear elliptic equations

We deduce maximum principles for a class of fourth order nonlinear elliptic equations by using auxiliary functions containing the square of the second gradient of the solution of such equations. A priori bounds on various quantities of interest are obtained.

Boundedness of solutions for anisotropic degenerate elliptic equations with nonstandard growth

Under nonstandard growth conditions, following Moser's iteration technique, we prove boundedness of solutions for fourth order nonlinear elliptic equation in divergence form.

Sun–Wong type theorems for second order damped elliptic equations

The oscillation criteria of Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549–560] are extended to the forced second order damped elliptic differential equation with mixed nonlinearities ∑ i , j = 1 N D i [ a i j ( x ) D j y ] + ∑ i = 1 N b i ( x ) D i y + C ( x , y ) = e ( x ) . When N = 1 , our results include and improve those of Nasr [A.H. Nasr, Sufficient conditions for the forced super-linear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc. 126 (1998) 123–125] for a forced second order superlinear differential equation, Wong [J.S.W. Wong, Oscillation criteria for a forced second order linear differential equations, J. Math. Anal. Appl. 231 (1999) 235–240] for a forced second order linear differential equation, Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549–560] for a forced second order differential equation with mixed nonlinearities. When N ⩾ 2 , our results are more general, and improve the oscillation criteria of Zhuang [R.-K. Zhuang, Annual oscillation criteria for second order nonlinear elliptic differential equations, J. Comput. Anal. Math. 217 (2008) 268–276] for a forced second order undamped elliptic differential equation.

On a Fourth Order Nonlinear Elliptic Equation with Negative Exponent

We consider the following nonlinear fourth order equation: $T\Delta u-D\Delta^2u=\frac{\lambda}{(L+u)^2}$, $-L<u<0$, in $\Omega$, $u=0$, $\Delta u=0$ on $\partial\Omega$, where $\lambda>0$ 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.

On efficient simulation of non-Newtonian flow in saturated porous media with a multigrid adaptive refinement solver

Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Here we discuss the efficient solution of the resulting second order nonlinear elliptic equation. The equation is discretized by the finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. We develop a special implementation, that allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here as an essential part of an accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid-adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.

Oscillation criteria for second order forced elliptic differential equations with mixed nonlinearities

In the case of oscillatory potentials, we establish some oscillation criteria for the second order forced elliptic differential equation with mixed nonlinearities div ( A ( x ) ∇ y ) + 〈 b ( x ) , ∇ y 〉 + C ( x , y ) = e ( x ) . When N = 1 , our results extend those of Kong [Q. Kong, Interval criteria for oscillation of second order linear ordinary differential equations, J. Math. Anal. Appl. 229 (1999) 258–270] for certain second order linear ordinary differential equations, Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second order differential equations with mixed nonlinearities, Appl. Math. Comput. (2007), doi:10.10.1016/j.amc.2007.08.042 ] for certain second order differential equations with mixed nonlinearities, Yang [Q. Yang, Interval oscillation criteria for a second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49–64] for a forced second order superlinear ordinary differential equation. When N ≥ 2 , our theorems are more general, and improve the oscillation criteria of Zhuang [R.-K. Zhuang, Annual oscillation criteria for second order nonlinear elliptic differential equations, J. Comput. Anal. Math. (2007), doi:10.1016/j.cam.2007.06.031 ] for a forced second order undamped elliptic differential equation.

On second order damped elliptic oscillation

New oscillation theorems are established for the second order nonlinear damped elliptic differential equation: ∑ i , j = 1 N D i [ a i j ( x ) D j y ] + ∑ i = 1 N b i ( x ) D i y + C ( x , y ) = 0 . The criteria obtained here involve integral averages of the coefficients of the above equation and improve some known oscillation results in literature by removing the differentiability of b i ( x ) for all i.

Fite and Kamenev type oscillation criteria for second order elliptic equations

Fite and Kamenev type oscillation criteria for the second order nonlinear damped elliptic differential equation $$ \sum_{i,j=1}^N D_i[a_{ij}(x)D_j y] + \sum_{i=1}^Nb_i(x)D_iy+p(x)f(y)=0 $$ are obtained. Our results are extensions of those for ordinary di

Some new oscillation theorems for second order nonlinear elliptic equations with damping

In this paper, some new oscillation criteria are obtained for second order elliptic differential equations with damping of the form ∇ ⋅ ( A ( x ) ∇ y ) + B T ( x ) ∇ y + q ( x ) f ( y ) = 0 , x ∈ Ω , where Ω is an exterior domain in R N . These criteria are different from most known ones in the sense that they are based on a class of new functions of the form Φ ( r , s , l ) defined in the sequel.