Existence of the Eigenvalues for the Cone Degenerate p-Laplacian∗

Hua CHEN Yawei WEI

Abstract The present paper is concerned with the eigenvalue problem for cone degenerate p-Laplacian.First the authors introduce the corresponding weighted Sobolev spaces with important inequalities and embedding properties.Then by adapting Lusternik-Schnirelman theory,they prove the existence of infinity many eigenvalues and eigenfunctions.Finally,the asymptotic behavior of the eigenvalues is given.

Keywords Quasi-linear,Degenerate operator,Variational methods

1 Introduction and Main Results

Then we have the following results.

2 Preliminaries

For the proof of the main result,the following inequalities and embeddings are necessary.

and it also induces an isomorphism

it follows that

3 Proof of Theorem 1.1

The idea of Lusternik-Schnirelman theory in[1]is adapted here for the proof.Consider the following two operators

Due to the assumption thatu1,u2are in a bounded set andδ=u1−u0,we have

which verifies the claim(3.3).

here,as in Lemma 3.1,we have

Recall the definition of duality map.

which implies thatT(u)is bounded onM.

where

and from(3.8)we have

which implies thatβk>0 for allk.Assume that there is no sequence inMverifying the conditions(3.13),then there must exist constantsδ >0 andρ>0 such that

According to the compact embedding stated in Lemma 2.1,it follows that up to a subsequence,

This shows(3.16).In the following,we verify that

In fact,

4 Proof of Corollary 1.1

AcknowledgementThe authors are grateful to the referees for their careful reading and valuable comments.