Landesman-Lazer conditions for strongly nonlinear boundary value problems
Lucio Boccardo, Pavel Drábek, Milan Kučera · Czech digital mathematics library · 1989
The solvability of the equations of the type div(|Vur»-2 Vu) + Ai|u| p -2 u + /(*,u) = g with Dirichlet or Neumann boundary conditions is proved for right-hand sides satisfying Landesman-Lazer type conditions.Here Ai is the smallest eigenvalue of the corresponding boundary value problem for the equation divflVul^Vu) + Xi |u| p -2 u = 0, p > 1, the growth of / is not greater then \u\ p ~l. Further, the solvability of the boundary value problem for the equation divflVup-2 Vu) + Ai|uj"~2u -|u|«" 2 u + /(*,u) = g for any right-hand side is proved under the assumption q > p.In both cases, a generalization to the equation with (p -l)-quasihomogeneous term with the second derivative is given.