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This article concerns with the global Holder regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Delta)^{s_1}_{p}+(-Delta)^{s_2}_{q}$, for $1<p,q<infty$ and $s_1,s_2in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior Holder regularity for local weak solutions, which need not be assumed bounded. The global Holder regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.
This paper is dedicated to the spectral optimization problem $$ mathrm{min}left{lambda_1^s(Omega)+cdots+lambda_m^s(Omega) + Lambda mathcal{L}_n(Omega)colon Omegasubset D mbox{ s-quasi-open}right} $$ where $Lambda>0, Dsubset mathbb{R}^n$ is a bounded
The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Delta)^{a
We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we i
In this paper, we deal with the existence and multiplicity of solutions for the fractional elliptic problems involving critical and supercritical Sobolev exponent via variational arguments. By means of the truncation combining with the Moser iteratio
It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $Phi$-Laplacian operator given by begin{equation*} left{ begin{array}{cl} displaystyle-Delta_Phi u= g(x,u), & mbox{in}~Omega, u=0,