On the natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva
On the necessity of gaps
Recent papers have studied the existence of phase transition solutions for Allen–Cahn type equations. These solutions are either single or multi-transition spatial heteroclinics or homoclinics between simpler equilibrium states. A sufficient condition for the construction of the multitransition solutions is that there are gaps in the ordered set of single transition solutions. In this paper we explore the necessity of these gap conditions.
On the Nehari manifold for a logarithmic fractional Schrödinger equation with possibly vanishing potentials
We study a class of logarithmic fractional Schrödinger equations with possibly vanishing potentials. By using the fibrering maps and the Nehari manifold we obtain the existence of at least one nontrivial solution.
On the Neumann problem for an elliptic system of equations involving the critical Sobolev exponent
We consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev nonlinearity. Our main objective is to study the effect of the coefficient of the critical Sobolev nonlinearity on the existence and nonexistence of least energy solutions. As a by-product we obtain a new weighted Sobolev inequality.
On the Neumann problem for systems of elliptic equations involving homogeneous nonlinearities of a critical degree
We establish the existence of solutions for the Neumann problem for a system of two equations involving a homogeneous nonlinearity of a critical degree. The existence of a solution is obtained by a constrained minimization with the aid of P.-L. Lions' concentration-compactness principle.
On the Neumann problem with combined nonlinearities
We establish the existence of multiple solutions of an asymptotically linear Neumann problem. These solutions are obtained via the mountain-pass principle and a local minimization.
On the Neumann problem with L¹ data
We investigate the solvability of the linear Neumann problem (1.1) with L¹ data. The results are applied to obtain existence theorems for a semilinear Neumann problem.
On the Newton-Kantorovich theorem and nonlinear finite element methods
Using a weaker version of the Newton-Kantorovich theorem, we provide a discretization result to find finite element solutions of elliptic boundary value problems. Our hypotheses are weaker and under the same computational cost lead to finer estimates on the distances involved and a more precise information on the location of the solution than before.
On the nodal line of the second eigenfunction of elliptic operators in two dimensions.
On the nodal lines of second eigenfunctions of the fixed membrane problem.
On the Nonexistence of a Hypersurface of Prescribed Mean Curvature with a Given Boundary.
On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity
We consider the Neumann problem for the equation , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues and . Applying a min-max principle based on topological linking we prove the existence of a solution.
On the nonlinear Neumann problem involving the critical Sobolev exponent and Hardy potential.
In this paper we investigate the solvability of some Neumann problems involving the critical Sobolev and Hardy exponents.
On the nonlinear Neumann problem with critical and supercritical nonlinearities [Book]
On the number of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds.
On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations
We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When has multiple critical points, (1.1) has a wide variety of positive solutions for small and the number of positive solutions increases to as . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.
On the number of positive solutions of some weakly nonlinear equations on annular regions.
On the numerical performance of a sharp a posteriori error estimator for some nonlinear elliptic problems
Karátson and Korotov developed a sharp upper global a posteriori error estimator for a large class of nonlinear problems of elliptic type, see J. Karátson, S. Korotov (2009). The goal of this paper is to check its numerical performance, and to demonstrate the efficiency and accuracy of this estimator on the base of quasilinear elliptic equations of the second order. The focus will be on the technical and numerical aspects and on the components of the error estimation, especially on the adequate...
On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation.
On the numerical solution of some semilinear elliptic problems.