Entire Large Solutions Research Articles

Existence and nonexistence of entire positive solutions for ( p , q ) -Laplacian elliptic system with a gradient term

This work is concerned with the entire positive solutions for a (p, q)-Laplacian elliptic system of equations with a gradient term. We find the sufficient condition for nonexistence of entire large positive solutions and existence of infinitely many entire solutions, which are large or bounded.

Entire large solutions for semilinear elliptic equations

We analyze the semilinear elliptic equation Δu=ρ(x)f(u), u>0 in RD(D⩾3), with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions u such that lim|x|→+∞u(x)=+∞. Assuming that f satisfies the Keller–Osserman growth assumption and that ρ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.

Anisotropic entire large solutions

Given q ∈ ( 0 , 1 ] , we construct nonradial entire large solutions to the equation Δ u = u q in R N .

Entire large solutions to semilinear elliptic systems

We consider the problem of existence of positive solutions to the elliptic system Δ u = p ( | x | ) v α , Δ v = q ( | x | ) u β on R n ( n ⩾ 3 ) which satisfies lim | x | → ∞ u ( x ) = lim | x | → ∞ v ( x ) = ∞ . The parameters α and β are positive, and the nonnegative functions p and q are continuous and min { p ( r ) , q ( r ) } does not have compact support. We show that if α β ⩽ 1 , then such a solution exists if and only if the functions p and q satisfy ∫ 0 ∞ t p ( t ) ( t 2 − n ∫ 0 t s n − 3 Q ( s ) d s ) α d t = ∞ , ∫ 0 ∞ t q ( t ) ( t 2 − n ∫ 0 t s n − 3 P ( s ) d s ) β d t = ∞ with P ( r ) = ∫ 0 r τ p ( τ ) d τ and Q ( r ) = ∫ 0 r τ q ( τ ) d τ . For α β > 1 , we show that a solution exists if either of the above conditions fails to hold; i.e., one of the integrals is finite. These extend all known results for the given system.

Existence and Nonexistence of Entire Positive Solutions for a Class of Singular p-Laplacian Elliptic System

In this paper, we show the existence and nonexistence of entire positive solutions for a class of singular elliptic system We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity. However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

Large solutions of coupled sublinear/superlinear elliptic equations

We show that large positive solutions exist for the semilinear elliptic equation Δu = p(x)u α + q(x)v β on bounded domains in R n , n ≥ 3, for the superlinear case 0 < α ≤ β, β > 1, but not the sublinear case 0 < α ≤ β ≤ 1. We also show that entire large positive solutions exist for both the superlinear and sublinear cases provided the nonnegative continuous functions p and q satisfy certain decay conditions at infinity. Existence and nonexistence of entire bounded solutions are established as well.

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term

We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term Δ u + | ∇ u | = p ( | x | ) f ( u , v ) , Δ v + | ∇ v | = q ( | x | ) g ( u , v ) on R N , N ⩾ 3 , provided that nonlinearities f and g are positive and continuous, the potentials p and q are continuous, c-positive and satisfy appropriate growth conditions at infinity. We find that entire large positive solutions fail to exist if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

Entire Large Solutions of Quasilinear Elliptic Equations of Mixed Type

In this paper, the existence and nonexistence of nonnegative entire large solutions for the quasilinear elliptic equation are established, where , and are nondecreasing and vanish at the origin. The locally H lder continuous functions and are nonnegative. We extend results previously obtained for special cases of and g.

Entire large solutions of semilinear elliptic equations of mixed type

We prove existence and nonexistence of nonnegative entire largesolutions for the semilinear elliptic equation $\Delta u = p(x)f(u) +q(x)g(u)$ in which $f$ and $g$ are nondecreasing and vanish at theorigin. The locally Hölder continuous functions $p$ and $q$ arenonnegative. We extend results previously obtained for special cases of$f$ and $g$.

On weak solutions of nonlinear weighted p-Laplacian elliptic inequalities

In this paper we give sufficient conditions for the nonexistence of nonnegative nontrivial entire weak solutions of class W loc 1 , p ( R n ) ∩ C ( R n ) of p -Laplacian elliptic inequalities with possibly singular weights. In order to get the results a new Omori–Yau type principle is used. We complement our nonexistence results by establishing existence of infinitely many positive radial solutions each of which blows up at some finite R > 0 . Finally, a criterium for the existence of positive entire large radial solutions of class W loc 1 , p ( R n ) ∩ C ( R n ) is also established.

Large solutions of mixed sublinear/superlinear elliptic equations

We consider the equation Δ u = p ( x ) u α + q ( x ) u β on R N ( N ⩾ 3 ) where p, q are nonnegative continuous functions and 0 < α ⩽ β . We establish conditions sufficient to ensure the existence and nonexistence of nonnegative entire large solutions of the equation.

Erratum: Non-existence of Entire Solutions of Degenerate Elliptic Inequalities with Weights

Non-existence results for non-negative distribution entire solutions of singular quasilinear elliptic differential inequalities with weights are established. Such inequalities include the capillarity equation with varying gravitational field h, as well as the general p-Poisson equation of radiative cooling with varying heat conduction coefficient g and varying radiation coefficient h. Since we deal with inequalities and positive weights, it is not restrictive to assume h radially symmetric. Theorem 1 extends in several directions previous results and says that solely entire large solutions can exist, while Theorem 2 shows that in the p-Laplacian case positive entire solutions cannot exist. The results are based on some qualitative properties of independent interest.

Existence of entire large positive solutions of a semilinear elliptic system

We show that large positive radial solutions exist for the semilinear elliptic system Δ u= p(| x|) f( v), Δ v= q(| x|) g( u) on R N ( N⩾3) for nonnegative, nondecreasing continuous functions f and g. Provided the nonnegative functions p and q are continuous, C-positive and satisfy the decay conditions ∫ ∞ 0tp(t) dt<∞ and ∫ ∞ 0tq(t) dt<∞ , and f, g satisfy ∫ ∞ 1[∫ s 0f(t) dt] −(1/2) ds<∞ , ∫ ∞ 1[∫ s 0g(t) dt] −(1/2) ds<∞ .

Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems

We show that entire large positive radial solutions exist for the semilinear elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on RN, N⩾3, for positive α and β, provided that the nonnegative functions p and q are continuous, c-positive, and satisfy the decay conditions ∫∞0tp(t)dt<∞ and ∫∞0tq(t)dt<∞ for α and β greater than unity, and ∫∞0tp(t)dt=∞ and ∫∞0tq(t)dt=∞ if neither α nor β is greater than one.

Some existence and nonexistence theorems for solutions of degenerate parabolic equations : H. A. Levine and P. E. Sacks, Depatment of Mathematics, Iowa State University, Ames, Iowa 50011, USA

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.