Remarks on the Shape of Least-energy Solutions to a Semilinear Dirichlet Problem

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Volume 14, Issue 4

Zongming Guo

J. Part. Diff. Eq., 14 (2001), pp. 365-383.

Published online: 2001-11

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Abstract

Structure of least-energy solutions to singularly perturbed semilinear Dirichlet problem ε²Δu - u^α + g(u) = 0 in Ω,u = 0 on ∂Ω, Ω ⊂ ⋅R^N a bounded smooth domain, is precisely studied as ε → 0^+, for 0 < α < 1 and a superlinear, subcritical nonlinearity g(u). It is shown that there are many least-energy solutions for the problem and they are spike-layer solutions. Moreover, the measure of each spike-layer is estimated as ε → 0^+ .

Keywords

Least-energy solutions spike-layer solutions singularly perturbed semilinear Dirichlet problem nontrivial nonnegative solutions

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@Article{JPDE-14-365, author = {}, title = {Remarks on the Shape of Least-energy Solutions to a Semilinear Dirichlet Problem}, journal = {Journal of Partial Differential Equations}, year = {2001}, volume = {14}, number = {4}, pages = {365--383}, abstract = { Structure of least-energy solutions to singularly perturbed semilinear Dirichlet problem ε²Δu - u^α + g(u) = 0 in Ω,u = 0 on ∂Ω, Ω ⊂ ⋅R^N a bounded smooth domain, is precisely studied as ε → 0^+, for 0 < α < 1 and a superlinear, subcritical nonlinearity g(u). It is shown that there are many least-energy solutions for the problem and they are spike-layer solutions. Moreover, the measure of each spike-layer is estimated as ε → 0^+ .}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5490.html} }

TY - JOUR T1 - Remarks on the Shape of Least-energy Solutions to a Semilinear Dirichlet Problem JO - Journal of Partial Differential Equations VL - 4 SP - 365 EP - 383 PY - 2001 DA - 2001/11 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5490.html KW - Least-energy solutions KW - spike-layer solutions KW - singularly perturbed semilinear Dirichlet problem KW - nontrivial nonnegative solutions AB - Structure of least-energy solutions to singularly perturbed semilinear Dirichlet problem ε²Δu - u^α + g(u) = 0 in Ω,u = 0 on ∂Ω, Ω ⊂ ⋅R^N a bounded smooth domain, is precisely studied as ε → 0^+, for 0 < α < 1 and a superlinear, subcritical nonlinearity g(u). It is shown that there are many least-energy solutions for the problem and they are spike-layer solutions. Moreover, the measure of each spike-layer is estimated as ε → 0^+ .

Zongming Guo . (2019). Remarks on the Shape of Least-energy Solutions to a Semilinear Dirichlet Problem. Journal of Partial Differential Equations. 14 (4). 365-383. doi:

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