Multiple solutions for biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

Ruichang Pei,Jihui Zhang

doi:10.1186/s13661-015-0378-5

Abstract

The main purpose of this paper is to establish the existence of three nontrivial solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using the minimax method and Morse theory.

Highlights

Consider the following Navier boundary value problem: ⎧⎨ u + c u = f (x, u), in ;⎩u = u =, in ∂, ( )where is the biharmonic operator and is a bounded smooth domain in RN (N ≥ )
We notice that almost all the works mentioned above involve the nonlinear term f (x, u) of a subcritical growth, say, (SCP): there exist positive constants c and c and q ∈ (, p∗ – ) such that f (x, t) ≤ c + c |t|q for all t ∈ R and x ∈, where p∗ = N/(N – ) denotes the critical Sobolev exponent
It is easy to see that seeking weak solutions of problem ( ) is equivalent to finding nonzero critical points of the following functional on H ( ) ∩ H ( ):

Summary

Introduction

Pei and Zhang Boundary Value Problems (2015) 2015:115 that there exist two or three solutions for a more general nonlinearity g by the variational method. Zhang [ ] proved the existence of solutions for a more general nonlinearity f (x, u) under some weaker assumptions. Zhang and Li [ ] proved the existence of multiple nontrivial solutions by means of Morse theory and local linking.

Objectives

Results

Conclusion