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An eigenvalue problem for elliptic systems. (English) Zbl 0996.35048

From the introduction: Let \(\Omega\) be a bounded and open subset of \(\mathbb{R}^n\) and \(N\geq 1\). The goal is to study the following eigenvalue problem \[ -\text{div}(\nabla_\xi L(x,u,\nabla u))+\nabla_sL(x,u,\nabla u)= \lambda \nabla_s G(x,u),\;(u,\lambda)\in M\times\mathbb{R},\tag{P} \] on the submanifold \(M= \{u \in W_0^{1,p}(\Omega, \mathbb{R}^N):\int_\Omega G(x,u)dx=1\}\) of \(W_0^{1,p} (\Omega,\mathbb{R}^N)\). Because the considered functionals \(f:W_0^{1,p} (\Omega, \mathbb{R}^N) \to\mathbb{R}\) defined by \(f(u)=\int_\Omega L(x,u,\nabla u)dx\) are in general not locally Lipschitzian the author uses non-smooth critical point theory and the subdifferential for continuous functions recently introduced in [I. Campa and M. Degiovanni, Subdifferential calculus and nonsmooth critical point theory, SIAM J. Optim. 10, No. 4, 1020-1048 (2000; Zbl 1042.49018)]. To prove that problem (P) admits a nontrivial weak solution in \(M\times\mathbb{R}\) he restricts \(f\) to \(M\) and looks for constrained critical points.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J45 Systems of elliptic equations, general (MSC2000)
47J30 Variational methods involving nonlinear operators
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems

Citations:

Zbl 1042.49018
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