Regularity and multiplicity results for fractional $(p,q)$-Laplacian equations

Abstract

This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{equation*} (-Δ)^{s_1}_{p}u+\ba(-Δ)^{s_2}_{q}u = \la a(x)|u|^{δ-2}u+ b(x)|u|^{r-2} u,\; \text{ in }\; \Om, \; u=0 \text{ on } \mathbb{R}^n\setminus \Om, \end{equation*} where $\Om$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary, $1< \de \le q\leq p<r \leq p^{*}_{s_1}$, with $p^{*}_{s_1}=\ds \frac{np}{n-ps_1}$, $0<s_2<s_1<1$, $n>p{s}_1$ and $\la, \ba>0$ are parameters. Here $a\in L^{\frac{r}{r-\de}}(\Om)$ and $b\in L^{\infty}(\Om)$ are sign changing functions. We prove the $L^{\infty }$ estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case $(r<p_{s_1}^{\ast }).$ Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex-concave problem. In case of $\de=q$, we show the existence of solution.

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