Elliptic Equations In RN Research Articles

Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝ n , II

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in ℝn by employing the Li’s method of energy function.

Existence of solutions of certain quasilinear elliptic equations in ℝN without conditions at infinity

This paper deals with conditions for the existence of solutions of the equations $$ - \sum\limits_{i = 1}^n {D_i A_i } (x,u,Du) + A_0 (x,u) = f(x), x \in \mathbb{R}^n ,$$ considered in the whole space ℝn, n ≥ 2. The functions A i (x, u, ξ), i = 1,…, n, A 0(x, u), and f(x) can arbitrarily grow as |x| → ∞. These functions satisfy generalized conditions of the monotone operator theory in the arguments u ∈ ℝ and ξ ∈ ℝn. We prove the existence theorem for a solution u ∈ W 1,ploc (ℝn) under the condition p > n.

On a Class of Semilinear Elliptic Equations in Rn

We establish that for n⩾3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses separated positive entire solutions of infinite multiplicity, provided that a locally Hölder continuous function K⩾0 in Rn\\{0}, satisfies K(x)=O(∣x∣σ) at x=0 for some σ>−2, and K(x)=c∣x∣−2+O(∣x∣−n[log∣x∣]q) near ∞ for some constants c>0 and q>0. In the radial case K(x)=∣x∣l1+∣x∣τ with l>−2 and τ⩾0, or K(x)=∣x∣λ−2(1+∣x∣2)λ/2 with λ>0, we investigate separation phenomena of positive radial solutions, and show that if n and p are large enough, the equation possesses a positive radial solution with initial value α at 0 for each α>0 and a unique positive radial singular solution among which any two solutions do not intersect.

Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN

We consider positive solutions of −div(|x|−2a∇u)=|x|−bpup−1, u⩾0 in RN, where for N⩾2: a<(N−2)/2, a<b<a+1, and p=2N/(N−2(1+a−b)). Ground state solutions are the extremal functions of the Caffarelli–Kohn–Nirenberg inequalities [6]. In [10] the authors have observed symmetry breaking phenomena for ground state solutions in a subregion of the parameters. In this paper, we continue our study on the structure of bound state solutions and construct bound state solutions having prescribed symmetry.

Existence of solutions of strongly nonlinear elliptic equations in Rn

The paper is dedicated to the existence of local solutions of strongly nonlinear equations in RN and the Orlicz spaces framework is used.

Nonexistence results of positive solutions for semilinear elliptic equations in Rn

We consider the global properties of nonnegative solutions of the semilinear elliptic equations in the entire space. By employing Pohozaev identity in the entire space and the results concerning the asymptotic behavior of nonnegative solutions, we establish some theorems of Liouville type.

Multiple positive solutions of quasilinear elliptic equations in RN

Multiple positive solutions of quasilinear elliptic equations in RN

Uniqueness of positive radial solutions of semilinear elliptic equations in RN and séré's non—degeneracy condition

Uniqueness of positive radial solutions of semilinear elliptic equations in RN and séré's non—degeneracy condition

Multiple positive solutions for semilinear elliptic equations in ℝN involving critical exponents

Multiple positive solutions for semilinear elliptic equations in ℝN involving critical exponents

An interpolation method to characterize domains of generators of semigroups

We describe a method for characterizing the domains of generators of semigroups enjoying suitable smoothing properties. Among the applications, such a method allows to prove new Schauder type theorems for elliptic equations in ℝn and for parabolic equations in [0,T] x ℝn.

A quasilinear elliptic equation in ℝN

A quasilinear elliptic equation in ℝN of Hamilton-Jacobi-Bellman type is studied. An optimal criterion for uniqueness which involves only a lower bound on the functions is given. The unique solution in this class is identified as the value function of the associated stochastic control problem.

Positive radial solutions of fully nonlinear elliptic equations inR n

By the Schauder-Tychonoff fixed-point theorem, we investigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations inRn. We give some sufficient conditions to guarantee the existence of bounded and unbounded radial solutions and consider the nonexistence of positive solution inRn.

L∞ ESTIMATE FOR SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS IN RN

In this paper, we get the L∞ estimate for the solutions of the following elliptic equation in RN with limiting Sobolev exponent and inhomogeneous term {-∑t=1N∂∂xi(|∇u|p-2∂u∂xi)+c(x)|u|p-2u=f(x,u)+g(x),x∈RN,u∈W1,p(RN), N>p>1and we show that any solution u of the above equation satisfies

Multiple solutions of a semilinear elliptic equation in ℝN

In this paper, we are concerned with the existence of multiple solutions of−Δu+u=λb(x)|u|p−1u+c(x)|u|q−1uwhere 1 <p, q <N+2N−2 if N ≧ 3, 1 <p, q < + ∞ if N = 2, λ > 0.We obtain the existence of multiple solutions by using concentrations-compactness method and dual variational principle to establish the corresponding existence of critical points.

Nonlinear Elliptic Equations in RN without Growth Restrictions on the Data

We show existence and regularity of solutions in R N to nonlinear elliptic equations of the form −div A( x, Du) + g( x, u) = ƒ when ƒ is just a locally integrable function, under appropriate growth conditions on A and g but not on ƒ. Roughly speaking, in the model case −Δ p ( u) + | u| s−1 u = ƒ with p > 2 − (1/ N), existence of a nonnegative solution in R N is guaranteed for every nonnegative ƒ ∈ L 1 loc( R N ) if and only if s > p − 1.

Radial symmetry of positive solutions of nonlinear elliptic equations in Rn

(1993). Radial symmetry of positive solutions of nonlinear elliptic equations in Rn. Communications in Partial Differential Equations: Vol. 18, No. 5-6, pp. 1043-1054.

On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation

SynopsisThis paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in ℝn. The equation is derived from the blow-up problem in the non-linear Schrödinger equation.

Sublinear elliptic equations in ℝn

(1) - Au = p(x)u a in A n , n > 3 with 0 0, p not identically zero. We shall assume throughout the paper that p E L m We look for a solution u > 0, u not identically zero, so that, by the loc" strong maximum principle, if such a solution exists then u > 0 in in

Blow-up behavior of ground states of semilinear elliptic equations in Rn involving critical Sobolev exponents

Blow-up behavior of ground states of semilinear elliptic equations in Rn involving critical Sobolev exponents

Existence of singular solutions of semi-linear elliptic equation in Rn

Existence of singular solutions of semi-linear elliptic equation in Rn