Abstract
We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes
Under mild assumptions on the nonlinear term g, we show the \(L^\infty \) boundedness of any weak solution (either not changing sign or sign-changing) by the Moser iteration method. Moreover, when \(s\in (0, \frac{1}{2}]\), we obtain that the solution is unique and actually belongs to \(C^{1,\alpha }(\overline{\Omega })\) for any \(\alpha \in (0,1)\).
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Notes
As a technical remark, the additional presence of the Laplace operator will produce an extra term (that is the first term in (3.3)). We will check that this term is nonnegative by using a suitable integration by parts for the second-order generalized derivative, which will allow us to successfully complete the estimate produced by the full mixed order operator.
As a notation remark, the convention used here is that
$$\begin{aligned} \int _{A}\int _B f(x,y)\, dydx:= \int _{A}\left( \int _B f(x,y)\, dy\right) dx. \end{aligned}$$
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The authors would like to thank the anonymous referee for carefully reading the manuscript and the valuable comments and suggestions on it.
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Both X. Su and Y. Wei are supported by the National Natural Science Foundation of China (Grant no. 11971060, 11871242). Y. Wei is supported by Natural Science Foundation of Jilin Province (Grant no. 20200201248JC), and Scientific Research Project of Education Department of Jilin Province (Grant no. JJKH20220964KJ)
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Su, X., Valdinoci, E., Wei, Y. et al. Regularity results for solutions of mixed local and nonlocal elliptic equations. Math. Z. 302, 1855–1878 (2022). https://doi.org/10.1007/s00209-022-03132-2
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DOI: https://doi.org/10.1007/s00209-022-03132-2