# Quasilinear elliptic equations with Neumann boundary and singularity

Quasilinear elliptic equations with Neumann boundary and singularity Let Ω be a bounded domain with a smooth C 2 boundary in ℝ N (N ≥ 3), 0 ∈ $$\bar \Omega$$ , and n denote the unit outward normal to ∂Ω. We are concerned with the Neumann boundary problems: −div(|x| α |∇u| p−2∇u) =|x| β u p(α,β)−1 − λ|xβ γ u p−1, u(x) > 0, x ∈ Ω, ∂u/∂n = 0 on ∂Ω, where 1 < p < N and α < 0, β < 0 such that $$p(\alpha ,\beta ) \triangleq \frac{{p(N + \beta )}} {{N - p + \alpha }}$$ > p, γ > α−p. For various parameters α, β or γ, we establish certain existence results of the solutions in the case 0 ∈ Ω or 0 ∈ ∂Ω. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Quasilinear elliptic equations with Neumann boundary and singularity

, Volume 25 (4) – Sep 8, 2009
16 pages

/lp/springer-journals/quasilinear-elliptic-equations-with-neumann-boundary-and-singularity-oKYhTvqZcD
Publisher
Springer Journals
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Math Applications in Computer Science; Applications of Mathematics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-008-8818-y
Publisher site
See Article on Publisher Site

### Abstract

Let Ω be a bounded domain with a smooth C 2 boundary in ℝ N (N ≥ 3), 0 ∈ $$\bar \Omega$$ , and n denote the unit outward normal to ∂Ω. We are concerned with the Neumann boundary problems: −div(|x| α |∇u| p−2∇u) =|x| β u p(α,β)−1 − λ|xβ γ u p−1, u(x) > 0, x ∈ Ω, ∂u/∂n = 0 on ∂Ω, where 1 < p < N and α < 0, β < 0 such that $$p(\alpha ,\beta ) \triangleq \frac{{p(N + \beta )}} {{N - p + \alpha }}$$ > p, γ > α−p. For various parameters α, β or γ, we establish certain existence results of the solutions in the case 0 ∈ Ω or 0 ∈ ∂Ω.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Sep 8, 2009

### References

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