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

Zonghu 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 -