THE PARABOLIC PROBLEM WITH ASYMPTOTICAL LINEARITY

초록

Let $\Omega$ be a bounded open subset of $R^n$ with smooth boundary $\partial \Omega$. Let $L$ be 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 $a_{ij}(x)\in L^{\infty }(\Omega )$. Let $f:R\to R$ be a $C^1$ function and $h\in L^2(\Omega )$. 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 ${\varphi }_1>0$. We consider the multiplicity of the solutions of the following parabolic boundary value problem $D_{t}u=\Delta u+f(u)-s{\varphi }_1-h(x)\text{mbox}in\text{mbox}\Omega \times R,\text{eqno}(1.1)$ $u(x,t)=0,x\in \partial \Omega ,\text{mbox}t\in R,$ $u(x,t)=u(x,t+2\pi ),\text{mbox}in\text{mbox}\Omega \times R.$ We consider the case $f(u)=b{u}^{+}-a{u}^{-}$, i. e., $D_{t}u=Lu+b{u}^{+}-a{u}^{-}-s{\varphi }_1-h(x)\text{mbox}in\text{mbox}\Omega \times R,\text{eqno}(1.2)$ $u(x,t)=0,x\in \partial \Omega ,\text{mbox}t\in R,$ $u(x,t)=u(x,t+2\pi ),\text{mbox}in\text{mbox}\Omega \times R.$ The main results are as follows: \Thm {1.1} Assume that $-\infty <a<{\lambda }_1<b<{\lambda }_2$ and $s>0$. Then (1.1) has at least three periodic solutions. \eop For the proof of Theorem 1.1 we use the Leray-Schauder degree theory and the variational reduction method.