This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{ equation*} (-\Delta)^{ s_1}_{ p}u+\ba(-\Delta)^{ s_2}_{ q}u = \la a(x)|u|^{ \delta-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 H\"older regularity of the weak solutions of the above problem in the subcritical case $(r<p_{ s_1}^*).$ 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.
Categories: math.AP