On the Fučí k spectrum for the $p$-Laplacian

Abstract

We prove that the set $\big\{(\alpha,\beta)\in\mathbb R^2 : $ the problem $-\Delta_pu=\alpha (u^+)^{p-1}-\beta (u^-)^{p-1}$ in $ \Omega,$ $ u=0$ on $ \partial\Omega$ has a nontrivial solution $\,\,\, \} $ contains infinitely many curves which exist locally in the neighbourhood of suitable eigenvalues of the p-Laplacian operator.