## Bounded Weak Solutions to an Elliptic Equation with Lower Order Terms and Degenerate Coercivity

Li Zhongqing,1, Gao Wenjie2

 基金资助: 国家自然科学基金.  11401252贵州财经大学引进人才科研启动基金.  2018YJ26

Received: 2017-12-15

 Fund supported: the NSFC.  11401252the Scientific Research Foundation for the Introduction of Talent in GUFE.  2018YJ26

Abstract

A boundary value problem to a class of elliptic equations with lower order terms and degenerate coercivity is studied in this paper. With help of De Giorgi iterative technique and Boccardo-Brezis's test function, the L estimate to weak solutions of the problem is obtained. Based upon the uniform L bound, the existence of bounded solution is proved.

Keywords： Elliptic equations ; Degenerate coercivity ; Lower order terms ; L regularity

Li Zhongqing, Gao Wenjie. Bounded Weak Solutions to an Elliptic Equation with Lower Order Terms and Degenerate Coercivity. Acta Mathematica Scientia[J], 2019, 39(3): 529-534 doi:

## 1 问题的介绍

$$$\left\{\begin{array}{ll} -\mbox{div}\bigg[\frac{1}{(1+|u|)^\theta}\nabla u\bigg]+\mu u+g(x, u, \nabla u) = f, \quad &x\in\Omega, \\ u(x) = 0, &x\in\partial\Omega, \end{array}\right.$$$

Boccardo等[4]考虑了一个拟线性椭圆方程,其最简模型为

$$$(L1)-(R1) \geq\frac{2\lambda-\gamma}{\lambda^2} \int_\Omega|\nabla[{\rm e}^{\lambda|G_k(H(u_n))|}-1]|^2{\rm d}x.$$$

$\begin{eqnarray} (L2)&\geq& \mu\int_{A_{n, k}}|u_n|[{\rm e}^{\lambda|G_k(H(u_n))|}-1]^2{\rm d}x\\ &\geq& \mu\hat{k}_0\left\|{\rm e}^{\lambda|G_k(H(u_n))|}-1\right\|^2_{L^2(\Omega)}, \end{eqnarray}$

Hölder不等式, $|f_n|\leq|f|$, $(R2)$可估计如下

$\begin{eqnarray} (R2)&\leq& 2\int_{A_{n, k}}|f_n|[{\rm e}^{\lambda|G_k(H(u_n))|}-1]^2{\rm d}x +\int_{A_{n, k}}|f_n|{\rm d}x\\ &\leq&\overbrace{2\int_\Omega|f|[{\rm e}^{\lambda|G_k(H(u_n))|}-1]^2{\rm d}x}^{(R_{21})} +\|f\|_{L^m(\Omega)}|A_{n, k}|^{\frac{1}{m^\prime}}, \end{eqnarray}$

$\begin{eqnarray} (R_{21})&\leq& 2\|f\|_{L^m(\Omega)}\left\|{\rm e}^{\lambda|G_k(H(u_n))|}-1\right\|_{L^{2m^\prime}(\Omega)}^2\\ & \leq& 2\|f\|_{L^m(\Omega)} \left\|{\rm e}^{\lambda|G_k(H(u_n))|}-1\right\|_{L^{2^\ast}(\Omega)}^{2\alpha} \left\|{\rm e}^{\lambda|G_k(H(u_n))|}-1\right\|_{L^2(\Omega)}^{2(1-\alpha)} \\ & \leq& 2\|f\|_{L^m(\Omega)} \left[\epsilon\left\|{\rm e}^{\lambda|G_k(H(u_n))|}-1\right\|_{L^{2^\ast}(\Omega)}^{2} +\epsilon^{-\frac{\alpha}{1-\alpha}}\left\|{\rm e}^{\lambda|G_k(H(u_n))|}-1\right\|_{L^2(\Omega)}^2\right]\\ & \leq&\epsilon 2\|f\|_{L^m(\Omega)}S_N^2\left\|\nabla[{\rm e}^{\lambda|G_k(H(u_n))|}-1]\right\|_{L^{2}(\Omega)}^{2}\\ & &+\epsilon^{-\frac{\alpha}{1-\alpha}}2\|f\|_{L^m(\Omega)}\left\|{\rm e}^{\lambda|G_k(H(u_n))|}-1\right\|_{L^2(\Omega)}^2, \end{eqnarray}$

$\epsilon 2\|f\|_{L^m(\Omega)}S_N^2 = \frac{2\lambda-\gamma}{2\lambda^2}$,取$k_0$满足

$$$[k_0(1-\theta)+1]^{\frac{1}{1-\theta}}-1 = \widehat{k_0} >\frac{\left[\frac{2\lambda-\gamma}{4\lambda^2\|f\|_{L^m(\Omega)}S_N^2}\right]^{-\frac{\alpha}{1-\alpha}}2\|f\|_{L^m(\Omega)}}{\mu},$$$

$\begin{eqnarray} \overbrace{\frac{2\lambda-\gamma}{2\lambda^2} \int_\Omega|\nabla[{\rm e}^{\lambda|G_k(H(u_n))|}-1]|^2{\rm d}x}^{(L_{11})} \leq \|f\|_{L^m(\Omega)}|A_{n, k}|^{\frac{1}{m^\prime}}. \end{eqnarray}$

$\begin{eqnarray} (L_{11}) &\geq&\frac{2\lambda-\gamma}{2\lambda^2S^2_N} \left(\int_\Omega|{\rm e}^{\lambda|G_k(H(u_n))|}-1|^{2^\ast}{\rm d}x\right)^{\frac{2}{2^\ast}}\\ &\geq&\frac{2\lambda-\gamma}{2S^2_N} \left(\int_{A_{n, k}}|G_k(H(u_n))|^{2^\ast}{\rm d}x\right)^{\frac{2}{2^\ast}}. \end{eqnarray}$

$$$\int_{A_{n, k}}|G_k(H(u_n))|^{2^\ast}{\rm d}x \leq C(N, \gamma, \|f\|_{L^m(\Omega)})|A_{n, k}|^{\frac{2^\ast}{2m^\prime}}.$$$

$$$\int_{A_{n, k}}|G_k(H(u_n))|^{2^\ast}{\rm d}x \geq (h-k)^{2^\ast}|A_{n, h}|.$$$

$$$|A_{n, h}| \leq \frac{C(N, \gamma, \|f\|_{L^m(\Omega)})}{(h-k)^{2^\ast}} |A_{n, k}|^{\frac{2^\ast}{2m^\prime}}.$$$

## 3 定理1.1的证明:解的存在性

一旦$\{u_n\}_{n = 1}^\infty$的先验$L^\infty$界得到,就可以证明方程解的存在性.事实上,由命题2.1,不妨假设存在$M > 0$,使得

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