涉及拉东测度的p-Laplacian退化椭圆方程的不存在性结果

Non-Existent Results of the p-Laplacian Degenerate Elliptic Equation Involving the Ladon Measure

摘要: 在本文中，我们研究了一类非线性p-Laplacian退化椭圆方程解的不存在性：$\{\begin{array}{ll}-div\left(\frac{a\left(x,\nabla u\right)}{{\left(1+u\right)}^{\gamma \left(p-1\right)}}\right)+u=\mu \hfill & x\in \Omega ,\hfill \\ u=0\hfill & x\in \partial \Omega .\hfill \end{array}$ 其中$1<p<2$ ，$\gamma >1$ ，且$\mu$ 是一个非负拉东测度，集中于调和容量为零的集合E上。本文首先利用截断技巧构造逼近方程，并借助Lions的结果和估计，获得逼近解的存在性及估计，这些估计保证了通过逼近问题的解取极限，可以得到原问题的解。最后，借助控制收敛定理、弱下半连续性及其他相关方法，证明了在源项为集中于调和容量为零的集合的拉东测度时，原问题的解趋于零。

Abstract: In this paper we study the existence and regularity of solutions for a class of nonlinear degenerate elliptic equations $\{\begin{array}{ll}-div\left(\frac{a\left(x,\nabla u\right)}{{\left(1+u\right)}^{\gamma \left(p-1\right)}}\right)+u=\mu \hfill & x\in \Omega ,\hfill \\ u=0\hfill & x\in \partial \Omega .\hfill \end{array}$ where $1<p<2$ , $\gamma >1$ , and $\mu$ is a nonnegative Radon measure concentrated on a set of zero harmonic capacity. This paper first employs a truncation technique to construct an approximating equation. By utilizing Lions’ results and estimates, the existence and a priori bounds of the approximate solution are obtained. These estimates ensure that the solution to the original problem can be derived by taking the limit of the solutions to the approximating problems. Finally, using the controlled convergence theorem, weak lower semicontinuity, and other related methods, it is proven that when the source term is a Radon measure concentrated on a set of zero harmonic capacity, the solution to the original problem tends to zero.