On the limiting problems for two eigenvalue systems and variations

Abstract

Let $\Omega$ be a bounded, smooth domain. Supposing that $\alpha(p) + \beta(p) = p$, $\forall\, p \in \left(\frac{N}{s},\infty\right)$ and $\displaystyle\lim_{p \to \infty} \alpha(p)/{p} = \theta \in (0,1)$, we consider two systems for the fractional $p$-Laplacian and a variation on the first system. The first system is the following. $$\left\{\begin{array}{ll} (-\Delta_p)^{s}u(x) = \lambda \alpha(p) \vert u \vert^{\alpha(p)-2} u \vert v(x_0)\vert^{\beta(p)} & {\rm in} \ \ \Omega,\\ (-\Delta_p)^{t}v(x) = \lambda \beta(p) \left(\displaystyle\int_{\Omega}\vert u \vert^{\alpha(p)} d x\right) \vert v(x_0) \vert^{\beta(p)-2} v(x_0) \delta_{x_0} & {\rm in} \ \ \Omega,\\ u= v=0 & {\rm in} \ \mathbb{R}^N\setminus\Omega, \end{array}\right. $$ where $x_0$ is a point in $\overline{\Omega}$, $\lambda$ is a parameter, $0<s\leq t<1$, $\delta_x$ denotes the Dirac delta distribution centered at $x$ and $p>N/s$. A variation on this system is obtained by considering $x_0$ to be a point where the function $v$ attains its maximum. The second one is the system $$\left\{\begin{array}{ll} (-\Delta_p)^{s}u(x) = \lambda \alpha(p) \vert u(x_1) \vert^{\alpha(p)-2} u(x_1) \vert v(x_2) \vert^{\beta(p)} \delta_{x_1} & {\rm in} \ \ \Omega,\\ (-\Delta_p)^{t}v(x) = \lambda \beta(p) \vert u(x_1) \vert^{\alpha(p)} \vert v(x_2) \vert^{\beta(p)-2} v(x_2) \delta_{x_2} & {\rm in} \ \ \Omega,\\ u= v=0 & {\rm in} \ \mathbb{R}^N\setminus\Omega, \end{array}\right. $$ where $x_1,x_2\in \Omega$ are arbitrary, $x_1\neq x_2$. Although we not consider here, a variation similar to that on the first system can be solved by practically the same method we apply. We obtain solutions for the systems (including the variation on the first system) and consider the asymptotic behavior of these solutions as $p\to\infty$. We prove that they converge, in the viscosity sense, to solutions of problems on $u$ and $v$.