The solvability and regularity results for elliptic equations involving mixed local and nonlocal p-Laplacian

Let \(\Omega \subset \mathbb {R}^{N}\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ for \(N\ge 2\) $N\ge 2$ be a bounded \(C^{1}\) $C^1$ domain. For \(0<s<1<p<N\) $0<s<1<p<N$ , we consider an elliptic problem involving mixed local and nonlocal p-Laplacian \(\begin{aligned} \left\{ \begin{aligned} -\Delta _p u+(-\Delta )_p^s u&=f(x) \quad{} & {} \text { in } \Omega , \\ u&=0{} & {} \text{ in } \mathbb {R}^N \backslash \Omega , \end{aligned}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill -{\mathrm{\Delta }}_{p}u+{(-\mathrm{\Delta })}_p^{s}u& =f(x)\hfill & \hfill \text{in}\mathrm{\Omega },\\ \hfill u& =0\hfill & \hfill \text{in}{\mathbb{R}}^N\backslash \mathrm{\Omega },\end{array}\right)\end{array}$$ where \((-\Delta )_p^s u\) ${(-\mathrm{\Delta })}_p^{s}u$ is the fractional p-Laplacian. We make use of Stampacchia Lemma and Solution Obtained as Limit of Approximations (SOLA) approach to prove the solvability and regularity of weak solution and distributional solution under assumption of \(f(x)\in L^{m}(\Omega )\) $f(x)\in L^m(\mathrm{\Omega })$ with different ranges of \(m>1\) $m>1$ , respectively.