Entire Large Solutions Research Articles

A necessary and sufficient condition for the existence of entire large solutions to a [formula omitted]-Hessian system

In this paper, we consider the existence of entire large solutions for a k-Hessian system. By using an iterative technique and some priori estimates, a necessary and sufficient condition for the existence of entire large solutions is established. The condition is easy to verify and our results generalize and improve some existing results.

Existence and nonexistence of entire large solutions to a class of generalized quasilinear Schrödinger equations

This paper is concerned to the class of generalized quasilinear Schrödinger equations which have appeared from plasma physics, as well as high-power ultrashort laser in matter. By introducing a variable replacement and using some iterative methods, the existence and nonexistence of entire large solutions are studied.

Existence and Asymptotic Behaviour of Entire Large Solutions for k-Hessian Equations

Existence and Asymptotic Behaviour of Entire Large Solutions for k-Hessian Equations

Entire large solutions to the k-Hessian equations with weights: Existence, uniqueness and asymptotic behavior

This paper is concerned with the k-Hessian equation Sk(D2u)=b(x)f(u) in RN, where b∈C∞(RN) is positive in RN, f∈C∞(β,∞) is positive and increasing on (β,∞) with β∈R∪{−∞} and satisfies the so-called Keller-Osserman condition. We first establish the existence of entire strictly k-convex large solutions to the equation. And then by constructing a new class of Karamata functions, we investigate the exact asymptotic behavior of entire strictly k-convex large solutions at infinity. Finally, we prove the uniqueness of solutions.

Entire large positive radial symmetry solutions for combined quasilinear elliptic system

Entire large positive radial symmetry solutions for combined quasilinear elliptic system

Existence of entire radial large solutions for a class of Monge–Ampère type equations and systems

This paper is mainly concerned with existence of entire positive radial large solutions for a class of Monge–Ampère type equations: det D 2 u ( x ) − α Δ u = a ( | x | ) f ( u ) , x ∈ ℝ N , and systems: det D 2 u ( x ) − α Δ u = a ( | x | ) f ( v ) , x ∈ ℝ N , det D 2 v ( x ) − β Δ v = b ( | x | ) g ( u ) , x ∈ ℝ N , where detD2u is the so-called Monge–Ampère operator, Δ is the classical Laplace operator, N≥2, α,β are positive constants, f,g:[0,∞)→[0,∞) are continuous and nondecreasing, and a,b:ℝN→[0,∞) are continuous.

The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach

In this paper, we establish some new results for the existence and nonexistence of radial large positive solutions for the modified Schrödinger system with a nonconvex diffusion term. Our main tools are the successive iteration technique and dual approach. A necessary and sufficient condition for the existence of radial large positive solutions is also given, which improves and extends many previous work in this field of research.

Entire Large Solutions to Semilinear Elliptic Systems of Competitive Type

Entire Large Solutions to Semilinear Elliptic Systems of Competitive Type

Symmetric solutions for an elliptic partial differential equation that arises in stochastic production planning with production constraints

In this article we consider the question of the existence of positive symmetric solutions to the problems of the following type Δu=a(|x|)h(u)+b(|x|)g(u) for x∈RN, which are called entire large solutions. Here N ≥ 3, and we assume that a and b are nonnegative continuous spherically symmetric functions on RN. We extend results previously obtained for special cases of h and g and we will describe a real-world model in which such problems may arise.

Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

In this paper, we analyze the blow-up rates and uniqueness of entire large solutions to the equation Delta u=a(x)f(u)+ mu b(x) |nabla u|^{q}, xin mathbb{R}^{N} (Ngeq 3), where mu > 0, q > 0 and a, bin mathrm {C}^{alpha }_{mathrm{loc}}(mathbb{R}^{N}) (alpha in (0, 1)). The weight a is nonnegative, b is able to change sign in mathbb{R}^{N}, and fin C^{1}[0, infty ) is positive and nondecreasing on (0, infty ) and rapidly or regularly varying at infinity. Additionally, we investigate the uniqueness of entire large solutions.

Entire large solutions to semilinear elliptic equations with rapidly or regularly varying nonlinearities

In this paper, we first show some necessary and sufficient conditions to the rapidly and regularly varying functions defined in Appendix A; in particular, we find that the rapidly varying function classes ℜ∞ and G∞ given in Definitions A.4 and A.5, respectively, are not equivalent, i.e., ℜ∞⫋G∞. Thereafter, we establish the asymptotic behavior and uniqueness of entire large solutions to the semilinear elliptic equation Δu=b(x)f(u),x∈RN(N≥3), where b∈C(RN) is positive in RN, f∈C[0,∞) is positive and non-decreasing on (0,∞), and belongs to a more general class of functions. Additionally, we also study the asymptotic behavior and uniqueness of entire large solutions when f∈G∞∖ℜ∞.

The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach

In this paper, we establish some new results on the existence and nonexistence of radial large positive solutions for a modified Schrödinger system with a nonconvex diffusion term by a successive iteration technique and the dual approach. The necessary and sufficient condition for the existence of radial large positive solutions is established. Our results improve and extend many previous work in this field of research.

Entire large solutions of semilinear equations involving the fractional Laplacian

We investigate the existence and nonexistence of positive entire large solutions of a class of semilinear equations involving the fractional Laplacian. Sharp lower and upper bounds of the minimal solution, if it exists, are also given.

Asymptotic behavior and uniqueness of entire large solutions to a quasilinear elliptic equation

Asymptotic behavior and uniqueness of entire large solutions to a quasilinear elliptic equation

Asymptotic behavior of entire large solutions to semilinear elliptic equations

In this paper, by using Karamata regular variation theory, we study the exact asymptotic behavior of entire large solutions to Δu=b(x)f(u),x∈RN(N≥3), where b∈C(RN) is nonnegative and nontrivial in RN, f∈C([0,∞)) is positive and non-decreasing on (0,∞), which satisfies Keller–Osserman condition and is rapidly varying or regularly varying at infinity with index γ≥1.

The entire large solutions for a quasilinear Schrödinger elliptic equation by the dual approach

In this paper, we consider a Modified Nonlinear Schrödinger equation Δu−Δ(|u|2γ)|u|2γ−2u=q(x)F(u) on Rn(n≥3), where γ>12, q is a nonnegative continuous function on Rn, F is a continuous positive and non-decreasing function on [0,∞). We establish sufficient conditions for the existence and nonexistence of positive entire large solutions for the given equation by using the dual approach.

Entire large solutions for a class of Schrödinger systems with a nonlinear random operator

In this paper, we study the nonexistence and existence of radial large positive solutions for a class of Schrödinger systems with a nonlinear random operator under certain growth conditions. We also establish the necessary and sufficient conditions for the existence of radial large positive solutions by a successive iteration technique. The results of the paper improve and extend many previous work in this field of research.

Large and entire large solutions for a class of nonlinear problems

In this paper, our main purpose is to establish the existence of entire large positive solutions to the quasilinear equation Δpu=β(x)h(u). We give also necessary conditions for the existence and nonexistence of such solutions. We distinguish the cases when β is radial and when it is nonradial.

Existence and Structure of Entire Large Positive Radial Solutions for a Class of Quasilinear Elliptic Systems

In this paper, we consider the problem for the existence of the entire positive radial solutions of quasilinear elliptic system  div(A1(|∇u|)∇u) = F (|x|, u, v), x ∈ R , div(A2(|∇v|)∇v) = G(|x|, u, v), x ∈ R , lim |x|→∞ u(x) = +∞, lim |x|→∞ v(x) = +∞, x ∈ R . And, we also consider the following quasilinear elliptic system:  div(A1(|∇u|)∇u) = p(|x|)g(v), x ∈ R , div(A2(|∇v|)∇v) = q(|x|)f(u), x ∈ R , lim |x|→∞ u(x) = +∞, lim |x|→∞ v(x) = +∞, x ∈ R . Using the theory of ordinary differential equation, iteration method and comparison principle, which studied the existence and structure of positive entire large radial solutions. The main results of the present paper are new and extend the previously known results.

Entire Large Solutions to Elliptic Equations of Power Non-linearities with Variable Exponents

Abstract We give conditions on the variable exponent q that ensure the existence and nonexistence of a positive solution to the elliptic equation Δu = uq(x) on ℝN (N ≥ 3) which satisfies lim|x|→∞ u(x) = ∞. The nonnegative function q is required to be locally Hölder continuous on ℝN. We prove existence for q > 1 provided q(x) decays to unity rapidly as |x| → ∞. We treat the case q ≤ 1 as a special case of q − 1 changing signs and show that a solution exists provided q is asymptotically radial. In addition, we give an example to show that our results are nearly optimal.