Multiple solutions for resonant problems of the Robin $p$-Laplacian plus an indefinite potential

摘要

We study a nonlinear boundary value problem driven by the $p$-Laplacian plusan indefinite potential with Robin boundary condition. The reaction term is aCarath\'eodory function which is asymptotically resonant at $\pm\infty$ withrespect to a nonprincipal Ljusternik-Schnirelmann eigenvalue. Using variationalmethods, together with Morse theory and truncation-perturbation techniques, weshow that the problem has at least three nontrivial smooth solutions, two ofwhich have a fixed sign.