Unbalanced $(p,2)$-fractional problems with critical growth

Abstract

We study the existence, multiplicity and regularity results of non-negative solutions of following doubly nonlocal problem: $$ (P_\la) \left\{ \begin{array}{lr}\ds \quad (-\Delta)^{s_1}u+\ba (-\Delta)^{s_2}_{p}u = \la a(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^r}{|x-y|^{\mu}}~dy\right)|u|^{r-2} u \quad \text{in}\; \Om, \quad \quad\quad \quad u =0\quad \text{in} \quad \mb R^n\setminus \Om, \end{array} \right. $$ where $\Om\subset\mb R^n$ is a bounded domain with $C^2$ boundary $\pa\Om$, $0<s_2 < s_1<1$, $n> 2 s_1$, $1< q<p< 2$, $1<r \leq 2^{*}_{\mu}$ with $2^{*}_{\mu}=\frac{2n-\mu}{n-2s_1}$, $\la,\ba>0$ and $a\in L^{\frac{d}{d-q}}(\Om)$, for some $q<d<2^{*}_{s_1}:=\frac{2n}{n-2s_1}$, is a sign changing function. We prove that each nonnegative weak solution of $(P_\la)$ is bounded. Furthermore, we obtain some existence and multiplicity results using Nehari manifold method.