Existence of bounded positive solutions of quasi-linear elliptic equations

Existence of bounded positive solutions of a class of quasi-linear elliptic equation is obtained in exterior domain of R ^N , N ≥ 1. Firstly, by using fixed point theory, the existence theorem of a class of ordinary differential equation is established. Then, by constructing super-solution and sub-solution, the existence of bounded positive solutions of quasi-linear elliptic equation is given. The results of this article are new and extend previously known results.     

Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth

We consider a classical semilinear elliptic equation with Neumann boundary conditions on an annulus in R ^N . The nonlinear term is the product of a radially symmetric coefficient with a pure power. We prove that if the power is sufficiently large, the problem admits at least three distinct positive and radial solutions. In case the coefficient is constant, we show that none of the three solutions is constant. The methods are variational and are based on the study of a suitable limit problem.     

Trivial stationary solutions to the Kuramoto-Sivashinsky and certain nonlinear elliptic equations

We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in R and R² are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic problems in RN.     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equations

In this paper, we study the existence and the uniqueness of positive solution for the sublinear elliptic equation, −Δu+u=p|u|sgn(u)+f in RN, N?3, 0<p<1, f∈L²(RN), f>0 a.e. in RN. We show by applying a minimizing method on the Nehari manifold that this problem has a unique positive solution in H¹(RN)∩Lp+1(RN). We study its continuity in the perturbation parameter f at 0.     

Existence and multiplicity of positive solutions of semilinear elliptic equations in unbounded domains

We investigate the existence and the multiplicity of positive solutions for the semilinear elliptic equation −Δu+u=Q(x)|u|^p−2u in exterior domain which is very close to RN. The potential Q(x) tends to positive constant at infinity and may change sign.     

Existence of entire positive solutions for a critical system of nonlinear elliptic equations

In this paper, using the fibering method introduced by Pohozaev, we establish an existence of multiple nontrivial positive solutions for a system of nonlinear elliptic equations in R^N with lack of compactness studying the properties of Palais-Smale sequence of the energy functional associated with the system.     

Elliptic equations with absorption in a half-space

We give a necessary and sufficient condition, in the spirit of the classical works by Keller and Osserman, for the elliptic equation Δu = f (u) to have a solution in a half-space of R^N. The function f is supposed to be nondecreasing and nonnegative, and we are interested in solutions whose range is where f > 0. The possibility of obtaining such a necessary and sufficient condition has been an open question for a long time.     

Positive solutions of an elliptic partial differential equation on R

In this paper, we discuss the existence, nonexistence and uniqueness of positive solutions of a one-parameter family of elliptic partial differential equations on R^N (N>2). These equations are of interests in mathematical biology and Riemannian geometry. Our approach are based on variational arguments and comparison principles.     

Properties of positive solutions to an elliptic equation with negative exponent

In this paper, we study some quantitative properties of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics as well as differential geometry. We can get some useful local gradient estimate and L¹ lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.     

Singular quasilinear elliptic problems with indefinite weights and critical potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN , is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.     

Nonexistence of bounded energy solutions for a critical exponent problem

In this paper, we discuss positive solutions for certain weighted elliptic equations with critical Sobolev exponent in R^N. The weights depend on a positive parameter γ, which is allowed to increase to infinity. While for small values of γ solutions are completely classified, an attempt to such a classification is much more difficult for large values of the parameter. In the present work we prove the nonexistence of solutions with bounded energy as γ increases to infinity. We also prove a multiplicity result for high energy solutions.     

Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \mathbb Rⁿ{\mathbb R^n} . The solutions take values in ordered Banach spaces E, e.g. E=\mathbb R^N{E=\mathbb R^N} ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.     

Existence and regularity results for relaxed Dirichlet problems with measure data

We study the following relaxed Dirichlet problem $$\left\{ \begin{gathered} Lu + \mu u = vin\Omega , \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$ where Ω is a bounded open subset ofR ^N,Lu=?div(A?u) is an elliptic operator, μ is a positive Borel measure on Ω not charging polar sets, and v is a measure with bounded variation on Ω. We give a definition of solution for such a problem, and then prove existence and regularity results. As a consequence, the Green function for relaxed Dirichlet problems can be defined, and some of its properties are proved, including the standard representation formula for solutions.     

On ground state solutions for singular and semi-linear problems including super-linear terms at infinity

We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation −Δu=λf(x,u),x∈R^N, where f:R^N×(0,∞)→[0,∞) is a suitable function and λ>0 is a real parameter. This result completes the principal theorem of A. Mohammed [A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.11.080] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure will be used.     

Nonlinear fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.     

Hirzebruch Functional Equation: Classification of Solutions

The Hirzebruch functional equation is \(\sum\nolimits_{i = 1}^n {\prod\nolimits_{j \ne i} {(1/f({z_j} - {z_i})) = c} } \) with constant c and initial conditions f(0) = 0 and f'(0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n ≤ 6 in the class of meromorphic functions and in the class of series. Previously, such results have been known only for n ≤ 4. The Todd function is the function determining the two-parameter Todd genus (i.e., the χ_a,b-genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N. It gives a solution to the Hirzebruch functional equation for n divisible by N. A series corresponding to a meromorphic function f with parameters in U ? ?^k is a series with parameters in the Zariski closure of U in ?^k, such that for the parameters in U it coincides with the series expansion at zero of f. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for n = 5 corresponds either to the Todd function or to the elliptic function of level 5. (2) Any series solution of the Hirzebruch functional equation for n = 6 corresponds either to the Todd function or to the elliptic function of level 2, 3, or 6. This gives a complete classification of complex genera that are fiber multiplicative with respect to ?P^n?1 for n ≤ 6. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level N for N = 2,..., 6 in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in ?⁴.     

Nonnegative solutions of an elliptic equation with a singular potential

We study the behavior of nonnegative solutions of the Dirichlet problem for a linear elliptic equation with a singular potential in the ball B = B(0,R) ⊂ R ⁿ (n ≥ 3), R ≤ 1. We find an exact condition on the potential ensuring the existence or absence of a nonnegative solution of that problem.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

A nonexistence theorem for an equation with critical Sobolev exponent in the half space

In this paper we prove the nonexistence of positive solutions of the equation-Δu=u^2*-1 inR ₊ ^N with certain homogeneous mixed boundary conditions. The proof is based on a monotonicity theorem obtained using the moving plane methods and some recent results of Berestycki and Nirenberg (see [BN]). The nonexistence theorem is applied to improve a result of [GP] on the characterization of the critical levels of a functional related to some nonlinear elliptic problem with critical Sobolev exponent and mixed boundary conditions.