On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p-Laplacian, the indefinite nonlinearity, and depend on the real parameter $\lambda$. A special focus is made on the extreme value of Nehari manifold $\lambda^*$, which determines the threshold of applicability of Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where parameter $\lambda$ lies above the threshold $\lambda^*$ is obtained.