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

Abstract

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.

Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

“…Motivated by these works we shall consider both subcritical and critical cases assuming that α > 0 and β ∈ R. In the same way, we shall consider both periodic and asymptotically periodic case for the functions V and g. It is worthwhile to mention that semilinear elliptic problem with periodic and asymptotically periodic terms have been also considered in the literature, see [28,29]. For further results on this subject we refer the reader also to [27,25,13,30]. In these works it was proved existence of solutions for the problem (1) taking into account the variational framework assuming that α = 1 and β = 0 in the subcritical or critical case.…”