Non-convergent radial solutions of semilinear elliptic equations (original) (raw)

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.

On the radial solutions of a nonlinear singular elliptic equation

International Journal of Mathematical Analysis, 2015

We study the existence and asymptotic behavior near the origin of radial entire solutions of the singular elliptic equation ∆ p U + αU + βx • ∇U + |x| l |U | q−1 U = 0 in IR N where p > 2, q ≥ 1, N ≥ 1, α < 0, β < 0 and l < 0. The behavior and the existence of positive solutions depends strongly on the sign of (N − p) l + p(N − 1) p − 1 .

Behavior of positive radial solutions for quasilinear elliptic equations

Proceedings of the American Mathematical Society, 2000

We establish a necessary and sufficient condition so that positive radial solutions to − div ( A ( | ∇ u | ) ∇ u ) = f ( u ) , in B R ( 0 ) ∖ { 0 } , R > 0 , \begin{equation*} -\textrm {div} (A(|\nabla u|)\nabla u) = f(u),\quad \mbox {in}~~ B_{R}(0)\setminus \{0\},\ R>0, \end{equation*} having an isolated singularity at x = 0 x=0 , behave like a corresponding fundamental solution. Here, A : R ∖ { 0 } → R A:\mathbb R\setminus \{0\}\to \mathbb R and f : [ 0 , ∞ ) → [ 0 , ∞ ) f:[0,\infty )\to [0,\infty ) are continuous functions satisfying some mild growth restrictions.

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 .