Multiplicity Of Solutions For Problems Of P-harmonic Elliptic Equations With Navier Boundary Value

As we all know, the research of boundary value problems always plays an important role in the study of differential equations. In particular, existence and multiplicity of solutions for boundary value problems, as an significant problems, has been aroused wide inter-est. In this process, many work were completed, which significantly advanced the developments of differential equations.Coming from physics and engineering and having profound back-grounds, many models of elliptic equations aroused wide interest in recent years. Among these problems, the second order elliptic equa-tion and p-Laplacian equation From the seventies of last century so far, using variational method to study the existence and multiplicity for elliptic boundary value problems has been widespread concerned. The idea of this approach is to translates the solutions of equations to the critical point of cor-responding energy functional. Many scholars applied Minimax Prin-ciple to obtain the existence and multiplicity in the framework of variational. The classic work of Ambrosetti and Rabinowitz in [2] is the most famous result, in which they introduce the following classic condition:there existθ> 2 and s0> 0, such that (AR) holds:This assumption can ensure the functional have the mountain pass geometry and satisfy the Palais-Smale condition.Costa and Magalhaes [7] replace the (AR) byIn 2003, Willem and Zou [24] assume H(x,s) is increasing with s, and whereMiyagaki and Souto [22] then improve the above results by and decreasing when s< - s0.In this paper, we consider the following p-harmonic fourth-order elliptic equation with Navier type boundary value condition: where 1 is a bounded smooth domain and f(x,s) is a continuous function onΩ×R.p-harmonic fourth-order elliptic equation with Navier type bound-ary value has profound backgrounds. It occurs in beam theory, for example (see [4]),1. a beam with small deformation;2. a beam of a material which satisfies a nonlinear power-like stress and strain law;3. a beam with two-sided links which satisfies a nonlinear power-like elasticity law.Many work are devoted to these problems. In [3], using lower-upper solutions, the author give the existence of solutions.In [9], the authors study the following problem and they point out that there exists the least eigenvalueλ1, Furthermore, this eigenvalue is simple, positive and isolated, and the corresponding eigenfunction is positive.In [19], Using Morse theory and local linking, the existence of solutions be obtained.In this paper, we study p-harmonic fourth-order elliptic equa-tions with Navier-type boundary value condition (6) When the non-linearity is super p-linear, we obtain the multiplicity of solutions of the problems, without assuming the famous Ambrosetti-Rabinowitz condition.