Nonlinear biharmonic boundary value problem

Tacksun Jung,Q-Heung Choi

doi:10.1186/1687-2770-2014-30

Tacksun Jung, Q-Heung Choi

Open Access

https://doi.org/10.1186/1687-2770-2014-30

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Abstract

We consider the nonlinear biharmonic equation with variable coefficient and polynomial growth nonlinearity and Dirichlet boundary condition. We get two theorems. One theorem says that there exists at least one bounded solution under some condition. The other one says that there exist at least two solutions, one of which is a bounded solution and the other of which has a large norm under some condition. We obtain this result by the variational method, generalized mountain pass geometry and the critical point theory of the associated functional.

Highlights

Let be a bounded domain in Rn with smooth boundary ∂ and L ( ) be a square integrable function space defined on
In this paper we study the following nonlinear biharmonic equation with Dirichlet boundary condition:
In Section, we prove Theorem . by the variational method, the generalized mountain pass geometry and the critical point theory

Summary

Introduction

Let be a bounded domain in Rn with smooth boundary ∂ and L ( ) be a square integrable function space defined on. Let be the elliptic operator and be the biharmonic operator. In this paper we study the following nonlinear biharmonic equation with Dirichlet boundary condition:. We assume that g satisfies the following conditions: (g ) g ∈ C(R, R), (g ) there are constants a , a ≥ such that g(u) ≤ a + a |u|μ– , where. We note that (g ) implies the existence of the positive constants a , a , a such that μ ξ g(ξ ) + a. The real number ξ in the definition (g ) is not automatically nonnegative.

Thus we obtain the boundedness of

Moreover since

Assume that is not bounded and there exists an