On the radial solutions of a nonlinear singular elliptic equation (original) (raw)

Existence of radial solutions for quasilinear elliptic equations with singular nonlinearities

Advanced Nonlinear Studies

We prove the existence of radial solutions of the quasilinear elliptic equation div(A(|Du|)Du) + f (u) = 0 in R n , n > 1, where f is either negative or positive for small u > 0, possibly singular at u = 0, and growths subcritically for large u. Our proofs use only elementary arguments based on a variational identity. No differentiability assumptions are made on f .

Radial nonlinear elliptic problems with singular or vanishing potentials

arXiv (Cornell University), 2017

In this paper we prove existence of radial solutions for the nonlinear elliptic problem −div(A(|x|)∇u) + V (|x|)u = K(|x|)f (u) in R N , with suitable hypotheses on the radial potentials A, V, K. We first get compact embeddings of radial weighted Sobolev spaces into sum of weighted Lebesgue spaces, and then we apply standard variational techniques to get existence results.

Positive and oscillatory radial solutions of semilinear elliptic equations

Journal of Applied Mathematics and Stochastic Analysis, 1997

We prove that the nonlinear partial differential equationΔu+f(u)+g(|x|,u)=0, in ℝn,n≥3, withu(0)>0, wherefandgare continuous,f(u)>0andg(|x|,u)>0foru>0, andlimu→0+f(u)uq=B>0, for 1<q<n/(n−2), has no positive or eventually positive radial solutions. Forg(|x|,u)≡0, whenn/(n−2)≤q<(n+2)/(n−2)the same conclusion holds provided2F(u)≥(1−2/n)uf(u), whereF(u)=∫0uf(s)ds. We also discuss the behavior of the radial solutions forf(u)=u3+u5andf(u)=u4+u5inℝ3wheng(|x|,u)≡0.

Non-convergent radial solutions of semilinear elliptic equations

Asymptotic Analysis, 1994

Maier, S., Non-convergent radial solutions of semilinear elliptic equations, Asymptotic Analysis 8 (1994) 363-377. 363 We study solutions u = u(t) of an initial value problem for u" + (n-l)/t u' + feu) = 0, which have the additional property that the limit of u as t approaches infinity does not exist. Besides some examples, we give necessary conditions on the non-linearity f for the existence of such (non-convergent) solutions. One corollary of these investigations will be that, if f(u)u >0 for small lui '# 0 then every solution u(.;p) (with Ipl small enough) of the initial value problem (1.1), stated below, converges to zero as t-+ 00.

Structure of Positive Radial Solutions of Semilinear Elliptic Equations

Journal of Differential Equations, 1997

We study the positive radial solutions of a semilinear elliptic equation 2u+ f (u)=0, where f (u) has a supercritical growth order for small u>0 and a subcritical growth order for large u. By showing the uniqueness of positive solutions behaving like O(|x| 2&n) at infinity, we give an almost complete description for the structure of positive radial solutions. As a consequence, we also prove the uniqueness of positive solutions of the nonlinear Dirichlet problem for the equation in a finite ball.

Asymptotic behavior and existence of solutions for singular elliptic equations

arXiv (Cornell University), 2019

We study the asymptotic behavior, as γ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is −∆u = f (x) u γ in Ω, where Ω is an open, bounded subset of R N and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f : either strictly positive on every compactly contained subset of Ω or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to −∆v + |∇v| 2 v = f in Ω.

Boundary singularities of positive solutions of some nonlinear elliptic equations

Comptes Rendus Mathematique, 2007

We study the behavior near x0 of any positive solution of (E) −∆u = u q in Ω which vanishes on ∂Ω \ {x0}, where Ω ⊂ R N is a smooth domain, q ≥ (N + 1)/(N − 1) and x0 ∈ ∂Ω. Our results are based upon a priori estimates of solutions of (E) and existence, non-existence and uniqueness results for solutions of some nonlinear elliptic equations on the upper-half unit sphere. To cite this article: