Hardy–Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients

We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations.     

Existence results for critical singular elliptic systems

In this paper, we consider semilinear elliptic systems with both singular and critical growth terms in bounded domains. The existence of a nontrivial solution is obtained by variational methods.     

Elliptic systems involving the critical exponents and potentials

In this paper, we investigate a singular elliptic system, which involves the critical Sobolev exponent and multiple Hardy-type terms. By employing variational methods, the existence of its positive solutions is established. By the Moser iteration method, some asymptotic properties of its nontrivial solutions at the singular points are verified.     

The geometry of solutions to a segregation problem for nondivergence systems

Segregation systems and their singular perturbations arise in different areas: particle anihilation, population dynamics, material sciences. In this article we study the elliptic and parabolic limits of a nonvariational singularly perturbed problem. Existence and regularity properties of solutions and their limits are obtained.     

Singular nonhomogeneous quasilinear elliptic equations with a convection term

In this work we establish existence results for a class of nonhomogeneous and singular quasilinear elliptic equations involving a convection term. The gradient term makes the problem non variational, and in addition to this difficulty we have to handle the singular term with a sign changing nonlinearity. The proof of the results are made combining the sub‐super solution method, fixed point theorem, Leray–Schauder degree theory and comparison theorems.     

Superharmonic functions are locally renormalized solutions

We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.     

Optimal boundary regularity for nonlinear singular elliptic equations

In this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary. By a careful construction of super- and sub-solutions, we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument.     

Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

In this paper, we study the Dirichlet problem for a class of infinitely degenerate nonlinear elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy's inequality, the existence and regularity of multiple nontrivial solutions have been proved.     

Nonexistence of solutions for singular elliptic equations with a quadratic gradient term

This paper concerns the nonexistence of solutions for singular elliptic equations with a quadratic gradient term. The main results complement and partly extend some works by Arcoya et al. (2009) [1]. As a by-product of the main results, we fill in a gap in one of their works.     

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.     

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)     

Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms

We prove some symmetry property for equations with Hardy terms in cones, without any assumption at infinity. We also show symmetry property and nonexistence of entire solutions of some elliptic systems with Hardy weights.     

Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli

The existence of multiple positive solutions of systems of singular Hammerstein integral equations is studied, where the nonlinearities involved are allowed to have singularities in their second variables and satisfy weaker conditions involving the first eigenvalues of the corresponding linear Hammerstein integral operators. Such systems contain some mathematical models arising in science and engineering. Applications are given to the existence of multiple positive radial solutions of systems of semilinear singular elliptic equations in annuli on which, to the best of our knowledge, there has been little study.     

The existence and asymptotics of solutions for the Abelian Chern–Simons system with two Higgs fields and two gauge fields

In this paper, we study a system of elliptic equations in R² which arises from the self-dual equations for the Abelian Chern–Simons system with two Higgs fields and two gauge fields. We provide a new proof for the existence of topological solutions by constructing explicit supersolutions and subsolutions. We also study the asymptotic behavior of condensate solutions on the torus. It is shown that the maximal solutions converge uniformly to zero away from the vortex points, and the convergence rate is computed.     

The singular set of stationary solution for semilinear elliptic equations with supercritical growth

In this paper, we are concerned with the partial regularity of the stationary solution to semilinear elliptic equations. For almost all supercritical exponents, we improve the upper bounds of the Hausdorff dimension of the singular set, which are derived by Pacard, to the optimal ones.     

Homogenization of singular numbers for a non self-adjoint elliptic problem in a perforated domain

We consider the spectral problem for a non self-adjoint Dirichlet problem for a higher-order elliptic operator in a sequence of perforated domains. We establish the convergence of the singular numbers generated by the problem to the corresponding singular numbers generated by a limit problem of the same type but containing an additional term of capacity type.Research supported by the National Research Foundation of South Africa.     

Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate or singular when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such non-linear elliptic equations.     

Nehari's problem and competing species systems

We consider an optimal partition problem in N-dimensional domains related to a method introduced by Nehari [22]. We prove existence of the minimal partition and some extremality conditions. Moreover we show some connections between the variational problem, the behaviour of competing species systems with large interaction and changing sign solutions to elliptic superlinear equations.     

Some remarks on singular solutions of nonlinear elliptic equations. I

We study strong maximum principles for singular solutions of nonlinear elliptic and degenerate elliptic equations of second order. An application is given on symmetry of positive solutions in a punctured ball using the method of moving planes. Dedicated to Felix Browder on his 80th birthday     

On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001