Ground States for Reaction-Diffusion Equations with Spectrum Point Zero

Abstract

In this paper, we are concerned with the existence of ground state solution for the following reaction-diffusion equation

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u-\Delta _x u+b(t,x)\cdot \nabla _x u+V(x)u+\lambda v=g(t,x,v),\\ - \partial _t v-\Delta _x v-b(t,x)\cdot \nabla _x v+V(x)v+\lambda u=f(t,x,u),\end{array} \right. \end{aligned}$$

where $(t,x)\in {\mathbb {R}}\times {\mathbb {R}}^N, z=(u,v): {\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\times {\mathbb {R}},\ b=(b_1,b_2,...,b_N) \in C^1({\mathbb {R}}\times {\mathbb {R}}^N)$, $V\in C({\mathbb {R}}^N, {\mathbb {R}}), f, g\in C({\mathbb {R}}\times {\mathbb {R}}^N \times {\mathbb {R}}, {\mathbb {R}})$, all four are depending periodically on t and x. The resulting problem engages four major difficulties: one is that the associated functional is strongly indefinite, the second is that the working space for our case is only a Banach space, not a Hilbert space, due to 0 is a boundary point of the spectrum of operator. The third difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences. The last difficulty is due to the lack of the strict monotonicity condition, seeking a ground state solution on the Nehari-Pankov manifold is difficult which needs some new techniques and methods. These enable us to develop a direct approach and new tricks to overcome the above difficulties. We obtain ground state solutions of Nehari-Pankov type under weaker conditions on the nonlinearity based on non-Nehari manifold method developed recently. To our best knowledge, our theorems appear to be the first such results about ground state solutions of diffusion equations with spectrum point zero.