Fourth Order Nonlinear Elliptic Equation Research Articles

Global Solution for a Fourth-Order Nonlinear Elliptic Equation

Objective: In this work, the existence of a global solution for a fourth-order nonlinear elliptic equation with boundary conditions is studied. Method: Variational methods are applied to the proposed model, naturally using the Banach Fixed Point Contraction Theorem. Results and Discussion: The conclusion of the result obtained is that for the nonlinear elliptic equation of the fourth order they have equivalent norms in the conveniently defined spaces. On the other hand, the existence of a solution exists whenever Banach's fixed-point theorem can be applied. Implications of the research: Apply in the first university years the application of global solution for a fourth-order nonlinear elliptic equation in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of global solution for a fourth-order nonlinear elliptic equation.

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.

Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.

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.

Exact thresholds for global existence to the nonlinear beam equations with and without a damping

In this paper, we study the Cauchy problem for focusing nonlinear beam equations with and without a damping term. By constructing two pairs of invariant flows, we obtain the exact thresholds for the global existence and blow-up to the above equations in the sense that both thresholds are explicitly expressed by the L2-norm of the fourth-order nonlinear elliptic equation without any damping.

Positive supersolutions of fourth-order nonlinear elliptic equations: explicit estimates and Liouville theorems

In this paper, we consider positive supersolutions of the semilinear fourth-order problem { ( − Δ ) 2 u = ρ ( x ) f ( u ) in Ω , − Δ u > 0 in Ω , where Ω is a domain in R N (bounded or not), f : D f = [ 0 , a f ) → [ 0 , ∞ ) ( 0 < a f ⩽ + ∞ ) is a non-decreasing continuous function with f ( u ) > 0 for u > 0 and ρ : Ω → R is a positive function. Using a maximum principle-based argument, we give explicit estimates on positive supersolutions that can easily be applied to obtain Liouville-type results for positive supersolutions either in exterior domains, or in unbounded domains Ω with the property that sup x ∈ Ω ⁡ dist ( x , ∂ Ω ) = ∞ . In particular, we consider the above problem with f ( u ) = u p ( p > 0 ) and with different weights ρ ( x ) = | x | a , e a x 1 or x 1 m ( m is an even integer). Also, when f is convex and ρ : Ω → ( 0 , ∞ ) is smooth with Δ ( ρ ) > 0 , then under an extra condition between f and ρ we show that every positive supersolution u of this problem with u = 0 on ∂Ω (Ω bounded) satisfies the inequality − Δ u ≥ 2 ρ ( x ) F ( u ) for all x ∈ Ω , where F ( t ) : = ∫ 0 t ( f ( s ) − f ( 0 ) ) d s .

Bounded solutions of fourth-order nonlinear elliptic equations with convection terms

ABSTRACT We study a Dirichlet problem associated to a nonlinear fourth-order equation with a convective lower order term and with coefficients of principal part satisfying a strengthened coercivity condition. We obtain the existence of a weak solution and its Hölder's continuity.

Fourth-order nonlinear elliptic equations with lower order term and natural growth conditions

We prove the existence of weak solutions of the homogeneous Dirichlet problem related to a class of nonlinear elliptic equations whose prototype is ∑∣α∣=2Dα[∣D2u∣p−2Dαu]−∑∣α∣=1Dα[∣D1u∣q−2Dαu]+u[∣D1u∣q+∣D2u∣p]=f where Ω is an open bounded subset of RN (N≥3) with sufficiently smooth boundary, u:Ω→R is the unknown function, Dhu={Dαu:∣α∣=h}, ∣Dhu∣=[∑∣α∣=h∣Dαu∣2]12, for h=1,2, numbers p, q∈[2,N[ and f∈L1(Ω).

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.

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.

Nonlinear fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of R n . Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.

Refined asymptotics for entire solutions of a biharmonic equation with a supercritical nonlinearity

We consider positive classical radially symmetric solutions of the prototype of a fourth-order nonlinear elliptic equation, $$\Delta^2 u = u^p, \quad x\in\mathbb R^N, \quad(\star)$$ where N ≥ 13 and p ≥ pc. Here \({p_c > \frac{N+4}{N-4}}\) is a critical exponent that is known to mark a borderline in respect of the asymptotic behavior in this problem (Gazzola and Grunau in Math Ann 334:905–936, 2006). For \({p > \frac{N+4}{N-4}}\), all positive radial solutions of (⋆) approach an explicitly known singular solution u∞ in the sense that \({\frac{u(x)}{u_\infty(x)}\to 1}\) as |x| → ∞. If p pc, and a rate involving a logarithmic correction when p = pc. These results parallel those known for the second-order counterpart of (⋆) (Gui et al. in Comm Pure Appl Math 45:1153–1181, 1992), but the technique is completely different.

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.

Existence of conformal metrics with constant Q-curvature

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author.

Classification of solutions of certain fourth-order nonlinear elliptic equations in ℝ4

We consider the uniqueness of solutions of the equation ?2u = eu in four-dimensional Euclidean space. Our main result is that the solutions are all classical ones, provided that the energy of the solutions is finite and the diffusion of the solutions decays to zero at infinity. The method we used in this paper is known as the method of moving spheres.