Asymptotic behavior of large solutions to \(H\)-systems with perturbations. (English) Zbl 1116.35048

Takahashi, Futoshi

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 58, No. 3-4, 459-475 (2004).

Summary: Let \(\Omega\subset\mathbb R^2\) be a bounded smooth domain. In this paper, we study the existence and the asymptotic behavior of solutions to the following Dirichlet problem of \(H\)-systems with perturbations: \[ \begin{cases} \Delta u=2u_{x_1}\wedge u_{x_2}-\varepsilon u\quad\text{in }\Omega,\\ u|_{\partial\Omega}=0,\end{cases}\tag{\(P_\varepsilon\)} \] where \(u=(u^1,u^2,u^3)\in H_0^1(\Omega;\mathbb R^3)\) and \(\varepsilon\in\mathbb R\).
We prove that for \(\varepsilon\in(0,\lambda_1)\) (\(\lambda_1\) denotes the first eigenvalue of \(-\Delta\) acting on \(H_0^1(\Omega;\mathbb R^3)\)), \((P_\varepsilon)\) admits at least one solution \(\overline u_\varepsilon\not \equiv 0\), which blows up at exactly one point in \(\Omega\) as \(\varepsilon\to 0\). We also characterize the location of the blowup point as the minimum point of a certain function defined on \(\Omega\), and give the exact blowup rate of \(\|\nabla\overline u_\varepsilon\|_{L^\infty(\Omega)}\) as \(\varepsilon\to 0\).

MSC:

35J55	Systems of elliptic equations, boundary value problems (MSC2000)
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

\(H\)-systems with perturbations; Blow up point; Asymptotic behavior

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