Title: Existence and regularity of weak solutions for mixed local and nonlocal semilinear elliptic equations Show Chinese title

Title: 混合局部与非局部半线性椭圆方程弱解的存在性与正则性

Abstract: We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{% \begin{array}{ll} -\Delta u + (-\Delta)^{s} u+ a(x)\ u =f(x,u) & \hbox{in $\Omega$,} u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$} \end{array}% \right. \end{equation*} where $s \in (0,1)$, $\Omega \subset \mathbb{R}^{n}$ is a bounded domain, the coefficient $a$ is a function of $x$ and the subcritical nonlinearity $f(x,u)$ has superlinear growth at zero and infinity. We show the existence of a non-trivial weak solution by Linking Theorem and Mountain Pass Theorem respectively for $\lambda_{1} \leqslant 0$ and $\lambda_{1} > 0$, where $\lambda_{1}$ denotes the first eigenvalue of $-\Delta + (-\Delta)^{s} +a(x)$. In particular, adding a symmetric condition to $f$, we obtain infinitely many solutions via Fountain Theorem. Moreover, for the regularity part, we first prove the $L^{\infty}$-boundedness of weak solutions and then establish up to $C^{2, \alpha}$-regularity up to boundary. Show Chinese abstract

Abstract: 我们研究由通常的拉普拉斯算子和分数拉普拉斯算子混合驱动的半线性椭圆方程的狄利克雷问题的弱解的存在性、多重性和正则性结果 \begin{equation*} \left\{% \begin{array}{ll} -\Delta u + (-\Delta)^{s} u+ a(x)\ u =f(x,u) & \hbox{in $\Omega$,} u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$} \end{array}% \right. \end{equation*} 其中 $s \in (0,1)$, $\Omega \subset \mathbb{R}^{n}$是一个有界区域，系数 $a$是 $x$的函数，次临界非线性 $f(x,u)$在零和无穷处具有超线性增长。 我们分别通过链接定理和山路定理证明了$\lambda_{1} \leqslant 0$和$\lambda_{1} > 0$的非平凡弱解的存在性，其中$\lambda_{1}$表示$-\Delta + (-\Delta)^{s} +a(x)$的第一个特征值。特别地，对$f$添加对称条件后，我们通过泉定理得到了无限多解。此外，在正则性部分，我们首先证明了弱解的$L^{\infty}$有界性，然后建立了直到边界处的$C^{2, \alpha}$正则性。