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Volume 5, Issue 3
On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

Xue Ruying

J. Part. Diff. Eq.,5(1992),pp.61-71

Published online: 1992-05

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  • Abstract
In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.
  • Keywords

Critical point theory quasilinear elliptic equation mixed boundary condition isoperimetric constant

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@Article{JPDE-5-61, author = {Xue Ruying}, title = {On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions}, journal = {Journal of Partial Differential Equations}, year = {1992}, volume = {5}, number = {3}, pages = {61--71}, abstract = { In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5745.html} }
TY - JOUR T1 - On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions AU - Xue Ruying JO - Journal of Partial Differential Equations VL - 3 SP - 61 EP - 71 PY - 1992 DA - 1992/05 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5745.html KW - Critical point theory KW - quasilinear elliptic equation KW - mixed boundary condition KW - isoperimetric constant AB - In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.
Xue Ruying. (1992). On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions. Journal of Partial Differential Equations. 5 (3). 61-71. doi:
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