Positive Solutions Of Semilinear Equations Research Articles

Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems

We apply the “monotone separation of graphs” technique of L.A. Peletier and J. Serrin [L.A. Peletier, J. Serrin, Uniqueness of positive solutions of semilinear equations in R n , Arch. Ration. Mech. Anal. 81 (2) (1983) 181–197; L.A. Peletier, J. Serrin, Uniqueness of nonnegative solutions of semilinear equations in R n , J. Differential Equations 61 (3) (1986) 380–397], as developed further by L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202], to the question of exact multiplicity of positive solutions for a class of semilinear equations on a unit ball in R n . We also observe that using P. Pucci and J. Serrin [P. Pucci, J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (2) (1998) 501–528] improvement of a certain identity of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] produces a short proof of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] result on the uniqueness of positive solution of ( 1 < p , q < n + 2 n − 2 ) Δ u + u p + u q = 0 for | x | < 1 , u = 0 when | x | = 1 .

Existence and Properties of Multiple Positive Solutions for Semi-Linear Equations with Critical Exponents

Existence and Properties of Multiple Positive Solutions for Semi-Linear Equations with Critical Exponents

On upper bounds for positive solutions of semilinear equations

Suppose that E is a bounded domain of class C 2, λ in R d and L is a uniformly elliptic operator in E. The set U of all positive solutions of the equation Lu= ψ( u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions u Γ ⩽ w Γ . We also introduced a class of solutions u ν in 1–1 correspondence with a certain class N 0 of σ-finite measures ν on ∂E. With every u∈ U we associated a pair ( Γ, ν) where Γ is a Borel subset of ∂E and ν∈ N 0 . We called this pair the fine boundary trace of u and we denoted in tr( u). Let u⊕ v stand for the maximal solution dominated by u+ v. We say that u belongs to the class E L,ψ if the condition tr( u)=( Γ, ν) implies that u⩽ w Γ ⊕ u ν and we say that u belongs to E L,ψ ∗ if the condition tr( u)=( Γ, ν) implies that u⩾ u Γ ⊕ u ν . It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class E L,ψ ∗ contains all bounded domains. It follows from results of Mselati (Thése de Doctorat de l'Université Paris 6, 2002; C.R. Acad. Sci. Paris Sér. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C 4 belong to E Δ,ψ where Δ is the Laplacian and ψ( u)= u 2. [Mselati proved that, in his case, u Γ = w Γ and therefore the condition tr( u)=( Γ, ν) implies u= u Γ ⊕ u ν = w Γ ⊕ u ν .] By modifying Mselati's arguments, we extend his result to ψ( u)= u α with 1< α⩽2 and all bounded domains of class C 2, λ . We start from proving a general localization theorem: E∈ E L,ψ under broad assumptions on L, ψ if, for every y∈ ∂E there exists a domain E′∈ E L,ψ such that E′⊂ E and ∂E∩ ∂E′ contains a neighborhood of y in ∂E.

A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form −Δu=λh(x)H(u−a)uq+u2*−1 in RN, where λ, a>0 are parameters, h(x) is both nonnegative and integrable on RN, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0≤q<2*−1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0≤q≤1 and 1<q<2*−1.

Existence results for the positive solutions of semilinear equations on the Heisenberg group

Existence results for the positive solutions of semilinear equations on the Heisenberg group

Positive Solutions of Semilinear Equations in Cones

Positive Solutions of Semilinear Equations in Cones

On the positive solutions of semilinear equations $\Delta u+\lambda u-hu\sp p=0$ on the compact manifolds

In this paper, we study the existence, nonexistence, and uniqueness of positive solutions of semilinear equations Δu+λu-hu p =0 on compact Riemannian manifolds as well as on bounded smooth domains in R n with homogeneous Dirichlet or Neumann boundary conditions

Positive solutions of semilinear equations in cones

In this paper we consider the problem of finding a positive solution of the equation Δ u + | x | ν u ( n + 2 + 2 ν ) / ( n − 2 ) = 0 \Delta u + |x{|^\nu }{u^{(n + 2 + 2\nu )/(n - 2)}} = 0 in a cone C \mathcal {C} , with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that u ( x ) = o ( | x | 2 − n ) u(x) = o(|x{|^{2 - n}}) , as C ∋ x → ∞ \mathcal {C} \ni x \to \infty ). We will always assume that ν &gt; − 2 \nu &gt; - 2 . We show that the existence of a solution depends on the sign of ν \nu and also on the shape of the cone C \mathcal {C} .

Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent

We study the asymptotic behavior of positive solutions of semilinear equations with nearly critical nonlinearity. The solutions are shown to blow up at exactly one point. The exact rate and location of blowing up are studied.

Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on ?3

We study the positive solutions of a semilinear equation with a Coulomb potential on ℝ3. We give a new uniqueness theorem for the positive radial solutions of such an equation and we apply these results to the Thomas-Fermi-Dirac-von Weizsacker equation without electrostatic repulsion.

The set of positive solutions of semilinear equations in large balls

SynopsisThe exact number of positive solutions of Δu + f(u) = 0 on finite balls in ℝN is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝN (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝN. This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).

Uniqueness of positive solutions of semilinear equations in ℝn

Uniqueness of positive solutions of semilinear equations in ℝn