Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.     

On a singular Neumann problem for semilinear elliptic equations with critical Sobolev exponent and lower order terms

We investigate the solvability of the Neumann problem involving the critical Sobolev exponent, the Hardy potential and a nonlinear term of lower order. Lower order terms are allowed to interfere with the spectrum of the operator subject to the Neumann boundary conditions. Solutions are obtained via a min-max procedure based on the variational mountain-pass principle and topological linking.      

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent

We establish the existence of a solution to the variational inequality (the obstacle problem) (1.1) which involves the critical Sobolev exponent. This result is also extended to an obstacle problem with a lower order perturbation. Dedicated to Professor F. Browder on the occasion of his 80-th birthday     

Solutions of the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy-type term

In the present paper, the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions to the problem is obtained.     

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

Existence of solutions for a nonlinear elliptic problem of fourth order with weight

In this work we study the existence and regularity of solutions of the equation Δ _p ² u = λm|u| ^q?2 u with the boundary conditions of Navier in the case p ≠ q.     

On symmetric solutions of a singular elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of the nontrivial solutions of the following problem:

     

Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.     

The exterior Dirichlet problem for a class of fourth order elliptic equations

In some exterior domain G of the Euclidian p-space $\text{R}$^p the Dirichlet boundary value problem is considered for the equation (L + κ²)²u = f, where L is a uniformly elliptic operator and κ is a real number different from 0. It can be shown that each solution u of this equation splits into u = x_l?_lu₁ + u₂, where u₁ and u₂ satisfy Heimholte equations. Asymptotic conditions for u are formulated by imposing Sommerfeld radiation conditions on u₁ and u₂. If u₁ and u₂ are assumed to satisfy the same radiation condition, we prove a “Fredholm alternative theorem.” If u₁ and u₂ satisfy different radiation conditions, existence and uniqueness of the solution can be shown, provided the space dimension p is greater than 2.     

On some second order and fourth order elliptic overdetermined problems

This paper deals with various overdetermined problems. In Sections 2 and 3 we investigate two problems involving the electrostatic potential confined in a region of ³ bounded by two tubular conductors, and we conclude under appropriate assumptions that the only possible configuration of the two conductors is two coaxial circular cylinders. In Section 4 we prove a conjecture of Payne and Schaefer for a fourth order overdetermined problem.     

Positive solutions of an elliptic equation with negative exponent: Stability and critical power

We study positive solutions of the equation

     

On the first eigenvalue of a fourth order Steklov problem

We prove some results about the first Steklov eigenvalue d ₁ of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d ₁ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.