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

## TL;DR

This paper investigates the existence of positive solution branches for p-Laplacian boundary value problems at the critical parameter value where the Nehari manifold method's applicability threshold is reached.

## Contribution

It establishes the existence of two solution branches for parameters above the Nehari manifold threshold, extending understanding of solution structure at this critical point.

## Key findings

- Existence of two positive solution branches above the threshold parameter
- Identification of the threshold parameter $\lambda^*$ for the Nehari manifold method
- Analysis of solution continuation at the extreme value of the Nehari manifold

## Abstract

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.