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Volume 4, Issue 1
Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents

Wang Chuanfang, Xue Ruying

J. Part. Diff. Eq.,4(1991),pp.77-96

Published online: 1991-04

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  • Abstract
Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.
  • Keywords

Semilinear elliptic equation Sobolev exponent Critical value Critical point (P &#8226 S) condition

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COPYRIGHT: © Global Science Press

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@Article{JPDE-4-77, author = {Wang Chuanfang, Xue Ruying}, title = {Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents}, journal = {Journal of Partial Differential Equations}, year = {1991}, volume = {4}, number = {1}, pages = {77--96}, abstract = { Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5763.html} }
TY - JOUR T1 - Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents AU - Wang Chuanfang, Xue Ruying JO - Journal of Partial Differential Equations VL - 1 SP - 77 EP - 96 PY - 1991 DA - 1991/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5763.html KW - Semilinear elliptic equation KW - Sobolev exponent KW - Critical value KW - Critical point KW - (P • S) condition AB - Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.
Wang Chuanfang, Xue Ruying. (1991). Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents. Journal of Partial Differential Equations. 4 (1). 77-96. doi:
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