Existence and asymptotic behavior of positive solutions for semilinear fractional Navier boundary-value problems

Abstract

We study the existence, uniqueness, and asymptotic behavior of positive continuous solutions to the fractional Navier boundary-value problem $$\displaylines{ D^{\beta }(D^{\alpha }u)(x)=-p(x)u^{\sigma },\quad \in (0,1), \cr \lim_{x\to 0}x^{1-\beta }D^{\alpha}u(x)=0,\quad u(1)=0, }$$ where $\alpha ,\beta \in (0,1]$ such that $\alpha +\beta >1$, $D^{\beta }$ and $D^{\alpha }$ stand for the standard Riemann-Liouville fractional derivatives, $\sigma \in (-1,1)$ and p being a nonnegative continuous function in (0,1) that may be singular at x=0 and satisfies some conditions related to the Karamata regular variation theory. Our approach is based on the Schauder fixed point theorem.

Keywords

• Fractional Navier differential equations
• Dirichlet problem
• positive solution
• asymptotic behavior
• Schauder fixed point theorem