On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions

Lu Yang,Xuan Wang

doi:10.1186/1687-2770-2014-117

Abstract

In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is and the convex nonlinear term is with . By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as , here the explicit expression of is provided. MSC:35J35, 35J40, 35J65.

Highlights

In recent years, there has been extensive attention on semilinear second-order elliptic equations, ⎧⎨–Δu = gλ(x, u), in, ⎩u =, on ∂, ( . )here is a bounded smooth domain in RN (N ≥ ), gλ : × R → R and λ is a positive parameter; see [ – ] and the references therein
They proved that problem admits at least two positive solutions for λ sufficiently small
We introduce some notations

Summary

Introduction

They proved that problem admits at least two positive solutions for λ sufficiently small. We consider the Nehari minimization problem: for λ > , αλ( ) = inf Jλ(u) | u ∈ Mλ( ) , where Mλ( ) = {u ∈ H( )\{ } | Jλ(u), u = }.

Results

Conclusion