# Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Annales Polonici Mathematici (2013)

- Volume: 109, Issue: 1, page 93-107
- ISSN: 0066-2216

## Access Full Article

top## Abstract

top## How to cite

topZonghu Xiu, and Caisheng Chen. "Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition." Annales Polonici Mathematici 109.1 (2013): 93-107. <http://eudml.org/doc/280966>.

@article{ZonghuXiu2013,

abstract = {We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities:
⎧ $-div(|x|^\{-ap\} |∇u|^\{p-2\} ∇u) + h(x)|u|^\{p-2\}u = g(x)|u|^\{r-2\}u$, x ∈ Ω,
⎨
⎩ $|x|^\{-ap\}|∇u|^\{p-2\} ∂u/∂ν = λf(x)|u|^\{q-2\}u$, x ∈ ∂Ω,
where Ω is an exterior domain in $ℝ^N$, that is, $Ω = \{ℝ^N\}∖D$, where D is a bounded domain in $ℝ^N$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method, we give a sufficient condition under which the problem has no nontrivial nonnegative solutions.},

author = {Zonghu Xiu, Caisheng Chen},

journal = {Annales Polonici Mathematici},

keywords = {singular quasilinear elliptic problem; variational methods; test function; concave and convex nonlinearities},

language = {eng},

number = {1},

pages = {93-107},

title = {Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition},

url = {http://eudml.org/doc/280966},

volume = {109},

year = {2013},

}

TY - JOUR

AU - Zonghu Xiu

AU - Caisheng Chen

TI - Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

JO - Annales Polonici Mathematici

PY - 2013

VL - 109

IS - 1

SP - 93

EP - 107

AB - We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities:
⎧ $-div(|x|^{-ap} |∇u|^{p-2} ∇u) + h(x)|u|^{p-2}u = g(x)|u|^{r-2}u$, x ∈ Ω,
⎨
⎩ $|x|^{-ap}|∇u|^{p-2} ∂u/∂ν = λf(x)|u|^{q-2}u$, x ∈ ∂Ω,
where Ω is an exterior domain in $ℝ^N$, that is, $Ω = {ℝ^N}∖D$, where D is a bounded domain in $ℝ^N$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method, we give a sufficient condition under which the problem has no nontrivial nonnegative solutions.

LA - eng

KW - singular quasilinear elliptic problem; variational methods; test function; concave and convex nonlinearities

UR - http://eudml.org/doc/280966

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.