Global pointwise estimates for Green's matrix of second order elliptic systems

We establish global pointwise bounds for the Green's matrix for divergence form, second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the Green's matrix. In the scalar case, such an estimate is a consequence of De Giorgi-Moser-Nash theory and holds for equations with bounded measurable coefficients in arbitrary domains. In the vectorial case, one need to impose certain assumptions on the coefficients of the system as well as on domains to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.     

The proof of the Lane-Emden conjecture in four space dimensions

We partially solve a well-known conjecture about the nonexistence of positive entire solutions to elliptic systems of Lane-Emden type when the pair of exponents lies below the critical Sobolev hyperbola. Up to now, the conjecture had been proved for radial solutions, or in n?3 space dimensions, or in certain subregions below the critical hyperbola for n?4. We here establish the conjecture in four space dimensions and we obtain a new region of nonexistence for n?5. Our proof is based on a delicate combination involving Rellich-Pohozaev type identities, a comparison property between components via the maximum principle, Sobolev and interpolation inequalities on Sn−1, and feedback and measure arguments. Such Liouville-type nonexistence results have many applications in the study of nonvariational elliptic systems.     

Conformally invariant fully nonlinear elliptic equations and isolated singularities

We study properties of solutions with isolated singularities to general conformally invariant fully nonlinear elliptic equations of second order. The properties being studied include radial symmetry and monotonicity of solutions in the punctured Euclidean space and the asymptotic behavior of solutions in a punctured ball. Some results apply to more general situations including more general fully nonlinear elliptic equations of second order, and some have been used in a companion paper to establish comparison principles and Liouville type theorems for degenerate elliptic equations.     

On elliptic systems with discontinuous nonlinearities

In this paper we deal with elliptic systems with discontinuous nonlinearities. The discontinuous nonlinearities are assumed to satisfy quasimonotone conditions. We shall use the method of upper and lower solutions with fixed point theorems on increasing operators in ordered Banach spaces to show some existence theorems.     

On the boundedness of solutions for degenerate nonlinear elliptic equations

We establish the boundedness of solutions of Dirichlet Problem for a class of degenerate nonlinear elliptic equations. To prove the result we follow a modification of Moser's method.     

Sobolev type embedding and quasilinear elliptic equations with radial potentials

We study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials. The existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions.     

On a semilinear elliptic problem without (PS) condition

An existence result for semilinear elliptic problems whose associated functionals do not satisfy a Palais-Smale condition is proved. The nonlinearity of our problem fits none of the conditions in Ambrosetti and Rabinowitz (J. Funct. Anal. 14 (1973) 349), de Figueiredo et al. (J. Math. Pures Appl. 61 (1982) 41) and Gidas and Spruck (Comm. Part. Diff. Eq. 6 (1981) 883). Some truncation happens to be essential, and in the argument some new results on Liouville-type theorems are established.     

On nontrivial solutions of critical polyharmonic elliptic systems

In the present work, we consider elliptic systems involving polyharmonic operators and critical exponents. We discuss the existence and nonexistence of nontrivial solutions to these systems. Our theorems improve and/or extend the ones established by Bartsch and Guo [T. Bartsch, Y. Guo, Existence and nonexistence results for critical growth polyharmonic elliptic systems, J. Differential Equations 220 (2006) 531-543] in both aspects of spectral interaction and regularity of lower order perturbations.     

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.     

A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations

We consider nonlinear elliptic equations driven by the p-Laplacian differential operator. Using degree theoretic arguments based on the degree map for operators of type (S)₊ , we prove theorems on the existence of multiple nontrivial solutions of constant sign.     

Nonvariational problems with critical growth

In this paper, we develop new topological methods for handling nonvariational elliptic problems of critical growth. Our primary goal is to demonstrate how concentration compactness can be applied to achieve topological existence theorems in the nonvariational setting. Our methods apply to both semilinear single equations and systems whose nonlinearity is of critical type.     

The Green function estimates for strongly elliptic systems of second order

We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq {\mathbb{R}}^n, n \geq 3We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain , under the assumption that solutions of the system satisfy De Giorgi-Nash type local H?lder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.     

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou(existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters,Volume 50, December 2015, Pages 48–55).     

On general boundary-value problems for nonlinear elliptic equations of second order in a multiply connected plane domain

This paper deals with some general irregular oblique derivative problems for nonlinear uniformly elliptic equations of second order in a multiply connected plane domain. Firstly, we state the well-posedness of a new set of modified boundary conditions. Secondly, we verify the existence of solutions of the modified boundary-value problem for harmonic functions, and then prove the solvability of the modified problem for nonlinear elliptic equations, which includes the original boundary-value problem (i.e. boundary conditions without involving undertermined functions data). Here, mainly, the location of the zeros of analytic functions, a priori estimates for solutions and the continuity method are used in deriving all these results. Furthermore, the present approach and setting seems to be new and different from what has been employed before.The research was partially supported by a UPGC Grant of Hong Kong.     

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.     

Sufficient structure conditions for uniqueness of viscosity solutions of semilinear and quasilinear equations

In this article, we introduce a new approach for proving Maximum Principle type results for viscosity solutions of second-order, fully nonlinear possibly degenerate elliptic equations. This approach leads, in particular, to a better understanding of the conditions on the equation which are necessary to obtain such results. It allows us to provide new comparison results for semilinear and quasilinear equations.     

Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

Using a combination of several methods, such as variational methods, the sub and supersolutions method, comparison principles and a priori estimates, we study existence, multiplicity, and the behavior with respect to λ of positive solutions of p-Laplace equations of the form −Δpu=λh(x,u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.     

Carleson Type Theorems for Certain Convolution Operators

We prove mapping theorems for some convolution operators, acting from Sobolev type spaces in to Lorentz spaces defined on with a fractional-order Carleson measure. As an application of the major theorems, we give some a priori estimates for the solutions of certain elliptic equations.     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.