Positive Entire Solutions Research Articles

Existence of positive entire solutions of a semilinear p-Laplacian problem with a gradient term

In this paper, we study a semilinear p-Laplacian problem −Δpu+h(x)|∇u| q = b(x)g(u), u > 0, x ∈ R N , lim |x|→∞ u(x )= 0, where q ∈ (p −1,p), b, h ∈C α(R N ) for some α ∈ (0,1), h(x) 0, b(x) > 0,∀x ∈ R N , and g ∈ C 1 ((0,∞),(0,∞)) which may be singular at 0. Using a sub-supersolution argument and a perturbed argument, we obtain the existence of entire solutions to the problem. No monotonicity condition is imposed on the functions g(s) and g(s) sp−1 .

Fundamental solutions and Liouville type theorems for nonlinear integral operators

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu⩽0 and for Iu+up⩽0, p>1. We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.

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.

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.

Qualitative properties of entire radial solutions for a biharmonic equation with supercritical nonlinearity

We study some qualitative properties of entire positive radial solutions of the supercritical semilinear biharmonic equation: in It is known from a paper by Gazzola and Grunau that there is a critical value of for and that has a singular solution . We show that for or and , any regular positive radial entire solution of intersects with infinitely many times. On the other hand, if and , then for all . Moreover, the solutions are strictly ordered with respect to the initial value .

Existence and asymptotic behavior of radially symmetric ground states of quasi-linear singular elliptic equations

Existence and asymptotic behavior of entire positive solutions of a class of quasi-linear elliptic equation - div ( | ∇ u | p - 2 ∇ u ) = ρ ( x ) f ( u ) , x ∈ R N is obtained. Under several hypotheses on the ρ ( x ) and f ( r ) , we obtain the existence of positive entire solution. Asymptotic behavior is discussed by constructing an upper solution. The results of this paper is new and extend previously known results.

Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms

We establish results concerning non-existence and existence of entire positive solutions for the nonlinear elliptic problem { − Δ p u = a ( x ) u m + λ b ( x ) u n in R N , u > 0 in R N , u ( x ) ⟶ | x | → ∞ 0 , where − ∞ < m < p − 1 < n with 1 < p < N ; a , b ≥ 0 , a , b ≠ 0 are locally Hölder continuous functions and λ ≥ 0 is a real parameter. The main purpose of this paper, in short, is to complement the principal theorem of B. Xu and Z. Yang (Entire bounded solutions for a class of quasi-linear elliptic equations, Boundary Value Problems 2007, Art. ID 16407, 8 pp) showing existence and non-existence of solutions for the above problem for λ > 0 appropriately.

Existence and non-existence of entire positive solutions for quasilinear systems with singular and super-linear terms

We establish the results concerning existence and non-existence of entire positive so- lutions for the nonlinear elliptic systems ⎧ ⎩ −Δpu = a(x)u m +λc(x)v n , x ∈ R N , −Δqv = b(x)v l +θc(x)u n , x ∈ R N , u,v > 0, x ∈ R N and u → 0,v → 0a s|x |→ ∞, where 1 < p,q < N and λ,θ 0 are nonnegative parameters, a,b,c : R N → (0,∞) are locally Holder continuous functions not identically zero, and −∞<m< p−1,−∞<l <q−1,max{p− 1,q −1} < n. The main purpose of this paper is to extend the principal theorem of Xu and Yang in (23) which concerned single equation.

A remark on the existence of entire positive solutions for a class of semilinear elliptic systems

Under the simple conditions on f and g, we show that entire positive radial solutions exist for the semilinear elliptic system Δ u = p ( | x | ) f ( v ) , Δ v = q ( | x | ) g ( u ) , x ∈ R N , N ⩾ 3 , where the functions p , q : [ 0 , ∞ ) → [ 0 , ∞ ) are continuous.

On positive entire solutions of indefinite semilinear elliptic equations

We establish that the elliptic equation Δ u + K ( x ) u p + μ f ( x ) = 0 in R n has a continuum of positive entire solutions for small μ ⩾ 0 under suitable conditions on K, p and f. In particular, K behaves like | x | l at ∞ for some l ⩾ − 2 , but may change sign in a compact region. For given l > − 2 , there is a critical exponent p c = p c ( n , l ) > 1 in the sense that the result holds for p ⩾ p c and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l = − 2 , the result holds for any p > 1 while p c is decreasing to 1 as l decreases to −2.

On a semilinear elliptic equation in Hn

We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

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 S n − 1 , and feedback and measure arguments. Such Liouville-type nonexistence results have many applications in the study of nonvariational elliptic systems.

Existence of positive entire solutions of a semilinear elliptic problem with a gradient term

By a sub–supersolution argument and a perturbed argument, we show the existence of entire solutions to a semilinear elliptic problem − Δ u + h ( x ) | ∇ u | q = b ( x ) g ( u ) , u > 0 , x ∈ R N , lim | x | → ∞ u ( x ) = 0 , where q ∈ ( 1 , 2 ] , b , h ∈ C loc α ( R N ) for some α ∈ ( 0 , 1 ) , h ( x ) ≥ 0 , b ( x ) > 0 , ∀ x ∈ R N , and g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) which may be singular at 0. No monotonicity condition is imposed on the functions g ( s ) and g ( s ) / s .

The existence and non-existence of positive solutions to a singular quasilinear elliptic problem in [formula omitted

This paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation − Δ p u = ρ a ( x ) g ( u ) + λ b ( x ) f ( u ) in R N , 1 < p < N , N ≥ 3 , with u ( x ) → 0 as | x | → ∞ . Here either g or f (or both of them) are singular at 0 in the sense that g ( t ) , f ( t ) → ∞ as t → 0 . By using a perturbation method which eliminates the singularity on a ball with radius R and then let R tends to infinity, with help of the bounded super-solution of the original problem, we obtain the existence of a weak solution of the problem. The main results of this paper improve the corresponding results of [Dragos-Patru Covei, the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in R N , Nonlinear Anal. 69 (2008) 2615–2622; J.V.Goncalves, C.A.Santos, Singular elliptic problems: Existence, non-existence and boundary behavior, Nonlinear Anal. 66 (2007) 2078–2090].

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.

Nonexistence of positive weak solutions of elliptic inequalities

In this paper we give sufficient conditions for the nonexistence of positive entire weak solutions of coercive and anticoercive elliptic inequalities, both of the p -Laplacian and of the mean curvature type, depending also on u and x inside the divergence term, while a gradient factor is included on the right-hand side. In particular, to prove our theorems we use a technique developed by Mitidieri and Pohozaev in [E. Mitidieri, S.I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001) 1–362], which relies on the method of test functions without using comparison and maximum principles. Their approach is essentially based first on a priori estimates and on the derivation of an asymptotics for the a priori estimate. Finally nonexistence of a solution is proved by contradiction.

Existence of positive entire solutions for semilinear elliptic systems in the whole space

This work is devoted to the existence ofpositive entire solutions for semilinear elliptic systems. With theaid of a degree theory argument, we use the shooting method andPohozaev-type identity to show the existence of positive radialsolutions.

Existence of positive entire solutions of semilinear elliptic equations

For n ≥ 3 and p > 1 , the elliptic equation Δ u + K ( x ) u p + f ( x ) = 0 in R n possesses a positive entire solution under proper conditions on locally Hölder continuous functions K and f . After obtaining a priori estimates on decay of solutions near ∞ , we study the existence of positive solutions, and present necessary conditions of f for the existence.

Positive solutions of anisotropic Yamabe-type equations in $\mathbb {R}^n$

We study entire positive solutions to the partial dierential equation in R n xu + ( + 1) 2 |x| 2 yu = | x| 2 u n+2 n 2, where x 2 R 2 , y 2 R n 2 , n 3 and > 0. We classify positive solutions with second order derivatives satisfying a suitable growth near the set x = 0.

Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case

We consider the Dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients. Under such minimal regularity assumptions, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. As a special case of our main theorem, assuming the coefficients are time-periodic, we obtain a new result on the existence of a principal eigenvalue of an associated (time-periodic) parabolic eigenvalue problem. We also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time-periodic uniformly parabolic operators. Finally, we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity.