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

• Autorzy: Zonghu Xiu , Caisheng Chen
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 35J92: Quasilinear elliptic equations with p -Laplacian
• 35B38: Critical points

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2013
• Tom: 109
• Numer: 1
• Strony: 93-107

Daty

wydano

2013

Twórcy

autor

Zonghu Xiu

• College of Science, Hohai University, Nanjing 210098, P.R. China
• Science and Information College, Qingdao Agricultural University, Qingdao 266109, P.R. China

autor

Caisheng Chen

• College of Science, Hohai University, Nanjing 210098, P.R. China

Identyfikatory

DOI

10.4064/ap109-1-7