On the eigenvalues and the spectrum for a class of semilinear elliptic operators. (English) Zbl 0597.35094

Let \(\Omega\) be a bounded open subset in \(R^ n\) with smooth boundary \(\partial \Omega\), \[ Lu=-\sum D_ i(a_{ij}(x)D_ ju)+a_ 0(x)u \] be a formally self-adjoint elliptic operator in \(\Omega\) with measurable \(L^{\infty}\) coefficients, \(a_ 0\geq 0\), \(f: \Omega\) \(\times R\to R\) be a continuous, odd in u (i.e. \(f(x,-u)=-f(x,u)),\) sublinear (i.e. \(| f(x,u)| \leq a| u| +b)\) function.
The author considers the eigenvalue problem: \(Lu+f(x,u)=\mu u\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). Using the Lyusternik-Schnirelman theory he establishes for each \(\alpha >0\) the existence of infinitely many eigenfunctions \(u_ k(\alpha)\) satisfying \(\int u^ 2_ k(x)dx=\alpha^ 2\). He also proves that for the numbers N(\(\mu)\) of the eigenvalues \(\mu_ k(\alpha)\), which are less or equal to a given \(\mu\), the formula \(N(\mu)=\kappa \mu^{n/2}+O(\mu^{(n-1)/2} \ln \mu)\) holds, as in the linear case.
Reviewer: Yu.V.Egorov


35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations