Higher order spectrum and problems of non-resonance. (Spectre d’ordre supérieur et problèmes de non-résonance.) (French) Zbl 0880.35083

Anane, Aomar; Chakrone, Omar; Gossez, Jean-Pierre

C. R. Acad. Sci., Paris, Sér. I, Math. 325, No. 1, 33-36 (1997).

Summary: We introduce a “higher order” spectrum for the Laplacian \(\Delta\), and more generally for the \(p\)-Laplacian \(\Delta_p\). This spectrum is completely characterized in the linear case of the Laplacian. As an application, we study the existence of solution in \(W^{1,p}_0(\Omega)\) of the equation \(-\Delta_pu= f(x,u,\nabla u)+ h(x)\), when the function \(f\) satisfies some nonresonance condition with respect to that spectrum.

MSC:

35P05	General topics in linear spectral theory for PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

\(p\)-Laplacian; nonresonance condition