Asymptotics for a variational problem with critical growth and slightly positive Dirichlet data. (English) Zbl 1155.35015
Let \(\Omega\subset{\mathbb R}^N\) (\(N>2\)) be a bounded domain with smooth boundary. Denote \(p=(N+2)/(N-2)\) and let \(\varepsilon>0\).
This paper deals with the study of the variational problem
\[ S_{\gamma,\varepsilon}=\inf_{u\in {\mathcal A}_{\gamma,\varepsilon}}\int_ \Omega | \nabla u| ^2, \]
where
\[ {\mathcal A}_{\gamma,\varepsilon}= \left\{u\in H^1(\Omega);\;u-\varepsilon\in H^1_0(\Omega),\;\gamma=\int_\Omega | u| ^{p+1}>| \Omega | \varepsilon^{p+1}\right\}. \]
The main result of this paper establishes that the solution concentrates at exactly one interior point as \(\varepsilon\) tends to zero. The proof combines variational arguments with a Pohozaev-type identity.
This paper deals with the study of the variational problem
\[ S_{\gamma,\varepsilon}=\inf_{u\in {\mathcal A}_{\gamma,\varepsilon}}\int_ \Omega | \nabla u| ^2, \]
where
\[ {\mathcal A}_{\gamma,\varepsilon}= \left\{u\in H^1(\Omega);\;u-\varepsilon\in H^1_0(\Omega),\;\gamma=\int_\Omega | u| ^{p+1}>| \Omega | \varepsilon^{p+1}\right\}. \]
The main result of this paper establishes that the solution concentrates at exactly one interior point as \(\varepsilon\) tends to zero. The proof combines variational arguments with a Pohozaev-type identity.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 49J35 | Existence of solutions for minimax problems |
| 35J60 | Nonlinear elliptic equations |
Keywords:
semilinear elliptic equation; critical Sobolev exponent; Robin function; Pohozaev’s identityReferences:
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