Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

Ruichang Pei,Jihui Zhang

doi:10.1186/s13661-014-0162-y

Abstract

The main purpose of this paper is to establish the existence of two nontrivial solutions and the existence of infinitely many solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using a suitable version of the mountain pass theorem and the symmetric mountain pass theorem.

Highlights

Consider the following Navier boundary value problem: ⎧⎨ u(x) + c u = f (x, u), in ;⎩u = u =, in ∂, ( )where is the biharmonic operator and is a bounded smooth domain in RN (N ≥ )
Our work is to study problem ( ) when nonlinearity f does not satisfy the (AR) condition, i.e., for some θ > and γ >
By the proof of Theorem . , we prove that I(u) satisfies condition (C)c∗ (c∗ > )

Summary

Introduction

For problem ( ) when f (x, u) = bg(x, u), Micheletti and Pistoia [ ] proved 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. We noticed that almost all of works (see [ – ]) mentioned above involve the nonlinear term f (x, u) of a subcritical (polynomial) 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.

Objectives

Results

Conclusion