Existence of Weak Solutions of P-Laplacian Problem

Abstract

This project deals with the variational and the Nehari manifold method ,or by the Nehari hypothesis for the p-Laplacian equations in a bounded domain or in the whole space.Then a proof of the existence of the weak solutions of the given p-Laplacian problem is given under certain specific conditions. In this project ,a different approach is used to tackle the proposed p-Laplacian problem which is by the variational method by using the mountain pass theorem which is used to guarantee the existence of solutions of the non-linear partial differential equation. Few information about various topics which is required to solve the proposed problem like the Sobolev spaces, Sobolev embeddings, distribution theory,Sobolev inequalities etc, are provided to have better understanding of the given p-Laplacian problem .The hypothesis by Nehari along with the techniques to solve non-linear partial differential equations is given along with some of the theorem like the Lax-Milgram theoerem,mountain pass theorem and the Banach fixed point theorem.