An Eigenvalue Problem for a Quasilinear Elliptic Equation

✍ Scribed by Yuanji Cheng

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 781 KB
• Volume: 196
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19981960103

✦ Synopsis

In this paper, we are concerned with the following eigenvalue problem:

domain and -Ap is the degenerate p-Laplace operator with p > 1.

An interesting special m e is when f = ( P ( Z ) ~U I ~~-~U + ~( ~) I U ( Q ~-~U , 0 < q1 < q2. By using the suband supersolutions method and the variational method, we prove the existence of the solution of the problem under certain growth conditions for f(z,u). In the above special case, our results give the existence of at least one positive solution for q2 > p' -1, where p' is the critical Sobolev exponent, and the existence of at least two positive and two negative solutions for q1 < p -1 < qz < p* -1.

We also present a 1D example which has many positive solutions for certain interval of X and for a special @ue of X it has even infinitely many solutions.