Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation

New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth-order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.

Multiple solutions for a fourth-order elliptic equation of Kirchhoff type with variable exponent

In this paper, we consider a class of fourth-order elliptic equations of Kirchhoff type with variable exponent [Formula: see text] where [Formula: see text], [Formula: see text], is a smooth bounded domain, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is the operator of fourth-order called the [Formula: see text]-biharmonic operator, [Formula: see text] is the [Formula: see text]-Laplacian, [Formula: see text] is a log-Hölder continuous function and [Formula: see text] is a continuous function satisfying some certain conditions. A multiplicity result for the problem is obtained by using the mountain pass theorem and Ekeland’s variational principle provided [Formula: see text] is small enough.

Existence and multiplicity of positive solutions for a fourth-order elliptic equation

We prove existence and multiplicity of solutions for the problem$$\left\{ {\matrix{ {\Delta ^2u + \lambda \Delta u = \vert u \vert ^{2*-2u},{\rm in }\Omega ,} \hfill \hfill \hfill \hfill \cr {u,-\Delta u > 0,\quad {\rm in}\;\Omega ,\quad u = \Delta u = 0,\quad {\rm on}\;\partial \Omega ,} \cr } } \right.$$where$\Omega \subset {\open R}^N$,$N \ges 5$, is a bounded regular domain,$\lambda >0$and$2^*=2N/(N-4)$is the critical Sobolev exponent for the embedding of$W^{2,2}(\Omega )$into the Lebesgue spaces.

HOMOGENIZATION OF ELASTIC PLATE EQUATION∗

We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Antoni´c and Balenovi´c (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation.