Multiplicity result for the parabolic problem with asymptotic linearity

초록

Let $\Omega$ be a bounded open subset of $R^n$ with smooth boundary $\partial \Omega$. We consider the multiplicity of the solutions of the following parabolic boundary value problem $D_{t}u=Lu+b{u}^{+}-a{u}^{-}-s{\varphi }_1-h(x)\text{mbox}in\text{mbox}\Omega \times R,\text{eqno}(1.2)$ $u(x,t)=0,x\in \partial \Omega ,\text{mbox}t\in R,$ $u(x,t)=u(x,t+2\pi ),\text{mbox}in\text{mbox}\Omega \times R.$ Here $L$ is the self-adjoint strongly elliptic partial differential operator $L={\sum }_{1\le i,j\le n}\frac{\partial }{\partial x_i}{a}_{ij}(x)\frac{\partial }{\partial x_j},$ where $a_{ij}(x)\in L^{\infty }(\Omega )$. The eigenvalue problem $Lu+\lambda u=0$ in $\Omega$ with Dirichlet boundary condition has infinitely many eigenvalues ${\lambda }_k$ and the associated normalized eigenfunctions ${\varphi }_k$, $k=1,2,\dots ,$ with $0<{\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_k\to \infty$ and ${\varphi }_1>0$.