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

TL;DR

This paper investigates the existence, multiplicity, and regularity of non-negative solutions for a complex nonlocal fractional problem involving critical growth, using variational methods and Nehari manifold techniques.

Contribution

It introduces new results on the existence and multiplicity of solutions for a doubly nonlocal fractional problem with critical growth, extending previous work to more general fractional operators.

Findings

Solutions are proven to be bounded.

Existence of solutions is established for certain parameter ranges.

Multiple solutions are obtained via Nehari manifold methods.

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.