Boundary behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up

By the Karamata regular variation theory and constructing comparison function, we show the exact asymptotic behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up.     

Multiple solutions of an inhomogeneous Neumann problem for an elliptic system with critical Sobolev exponent

In this paper we prove the existence of two solutions for the inhomogeneous Neumann problem with critical Sobolev exponent.     

Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem

We investigate singular and degenerate behavior of solutions of the unstable free boundary problem

Δu=−χ{u>0}.     

Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions u decaying to zero at infinity, for a class of equations of Lane-Emden-Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ(u) where σ:(0,∞)→(0,∞) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.     

Existence and regularity of nonnegative solution of a singular quasi-linear anisotropic elliptic boundary value problem with gradient terms

In this paper, we consider the singular quasi-linear anisotropic elliptic boundary value problem

(P)     

Analytical methods for an elliptic singular perturbation problem in a circle

We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain.     

An inverse problem for an elliptic equation with an affine term

We consider the following elliptic boundary value problem: on , u = 0 on where is a smooth bounded planar domain. We show that for a large class of domains and for any such that is not identically constant there exist at most finitely many different pairs of coefficients such that the problem has a solution with the normal flux on . Received: 4 February 1999     

Boundary behaviour for solutions of elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet boundary value problem for a class of second-order nonlinear elliptic partial differential equations with a gradient term and singular data. Under general conditions on the data, we study the behaviour of the solution near the boundary of the domain. Under suitable additional conditions we also investigate the second-order term in the asymptotic expansion of the solution in terms of the distance from the boundary.     

A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system

An elliptic system is considered in a smooth bounded domain, subject to Dirichlet boundary conditions of three different types. Based on the construction of certain upper and sub-solutions, we obtain some conditions on the parameters ai,bi,ci (i=1,2) and the exponents m,n,p,q to ensure the existence of positive solutions. Furthermore, uniqueness and boundary behavior of positive solutions is also discussed.     

On radial solutions of certain semi-linear elliptic equations

We consider the semi-linear elliptic equation Δu+f(x,u)+g(|x|)x·∇u=0$\mathrm{\Delta }u+f(x,u)+g(|x|)x·\nabla u=0$, in some exterior region of Rⁿ,n?3${\mathbb{R}}^n,n?3$. It is shown that if f$f$ depends radially on its first argument and is nonincreasing in its second, boundary conditions force the unique solution to be radial. Under different conditions, we prove the existence of a positive radial asymptotic solution to the same equation.     

Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions

This work deals with the analysis of problems

     

Quasilinear elliptic system with coupling on nonhomogeneous critical term

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with the possibility of coupling on the critical and subcritical terms which are not necessarily homogeneous. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. A version of the Concentration-Compactness Principle for this class of systems allows us to verify that the Palais–Smale condition is satisfied below a certain level.     

Large solution to nonlinear elliptic equation with nonlinear gradient terms

The aim of this paper is to study the qualitative behavior of large solutions to the following problem

     

Superlinear elliptic equations with singular coefficients on the boundary

In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to .     

A compactly supported solution to a three-dimensional uniformly elliptic equation without zero-order term

A second order, nonvariational, elliptic operator L and a function V are constructed in with the following properties: the operator L is uniformly elliptic, without zero-order term and smooth almost everywhere in ; the function (1<p<3) solves the equation LV=0 in , it has compact support but it is not identically zero.     

Nonlinear elliptic equations with measure data

In this paper we prove the existence of solutions of nonlinear equations of the type-div(a(x, u, Du)+H(x, u, Du)=f, wherea andH are Caratheodory functions andf is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results.