High energy solutions for the fourth-order elliptic equations in R N

Bitao Cheng

doi:10.1186/s13661-014-0199-y

Abstract

In the present paper, we study the following fourth-order elliptic equations: , for , , where , . Under more relaxed assumptions on , by using some special techniques, a new existence result of high energy solutions is obtained via the symmetric mountain pass theorem.

Highlights

Introduction and main resultsIn this paper, we consider the following fourth-order elliptic equations:u – u + V (x)u = f (x, u), for x ∈ RN, u(x) ∈ H (RN ), ( . )where := ( ) is the biharmonic operator, V ∈ C(RN, R), f ∈ C(RN × R, R) satisfy some further conditions.When is a smooth bounded domain in RN, the problem u + c u = f (x, u), in,u = u =, on ∂, arises in many applications from mathematical physics, which is usually used to describe some phenomena appearing in various physical, engineering, and other sciences
We present a new proof technique to verify the boundedness of Palais-Smale sequences, and we apply the classical symmetric mountain pass theorem of Rabinowitz instead of the fountain theorem in [ ] to obtain high energy solutions for the problem ( . )
Competing interests The author declares that he has no competing interests

Summary

There exist μ

ΜF(x, t) – f (x, t)t ≤ a|t| , for a.e. x ∈ RN and ∀|t| ≥ L. We can state our result about the existence of a sequence of high energy solutions for the fourth-order elliptic equations ) possesses a sequence of high energy solutions in E. Conditions (f )-(f ) are much weaker than (f )-(f ). (f )-(f ) are much weaker than (f )-(f ). Recall that we say I satisfies the (PS) condition at the level c ∈ R ((PS)c condition for short) if any sequence {un} ⊂ E along with I(un) → c and I (un) → as n → ∞ possesses a convergent subsequence. Let assumption (V ) and (f ) hold. Any bounded Palais-Smale sequence of I has a strongly convergent subsequence in E. Proof Let {un} ⊂ E be any bounded Palais-Smale sequence of I.

Since the embedding

An μV