Asymptotic properties of variational eigenvalues for semilinear elliptic operators. (English) Zbl 0689.35070

Shibata, Tetsutaro

Boll. Unione Mat. Ital, VII. Ser., B 2, No. 2, 411-425 (1988).

The purpose of this paper is to characterize the accumulation points of the n-th variational eigenvalues of the following semilinear eigenvalue problem: \[ Lu+f(x,u)=\mu u\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] where \(\Omega \subset {\mathbb{R}}^ N(N\geq 3)\) is a bounded domain with a smooth boundary \(\partial \Omega\) and \[ Lu:=- \sum^{N}_{i,j=1}\partial_{x_ j}(a_{ij}(x)\partial_{x_ i}u)+a_ 0(x)u \] is a formally self adjoint operator with \(L^{\infty}\) coefficients such that \(a_{ij}=a_{ji}(i,j=1,2,...,N)\) and \(a_ 0\geq 0\), which is uniformly elliptic in \(\Omega\).

MSC:

35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

semilinear elliptic; eigenvalue problem; asymptotic properties; variational eigenvalues; semilinear elliptic operators