Existence of solutions for fractional $p\&q$-Kirchhoff system in unbounded domain
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In this paper, we investigate the fractional $p\&q$-Kirchhoff type system $$\left\{ \begin{array}{l} M_1([u]^p_{s,p})(-\Delta)_p^su+V_1(x)|u|^{p-2}u\\ \quad=\ell k^{-1}F_u(x,u,v)+\lambda a(x)|u|^{m-2}u,\quad x\in\Omega,\\ M_2([u]^q_{s,q})(-\Delta)_q^sv+V_2(x)|v|^{q-2}v\\ \quad=\ell k^{-1}F_v(x,u,v)+\mu a(x)|v|^{m-2}v,\quad\ x\in\Omega,\\ u\,=v\,=0,\qquad\qquad\qquad\qquad\qquad\qquad\ \ x\in\partial\Omega, \end{array} \right.$$ where $\Omega\subset \mathbb{R}^N$ is an unbounded domain with smooth boundary $\partial\Omega$, and $0<s<1<p\leq q$ and $sq<N,$ $\lambda,\mu>0$, $1<m\leq k<p_s^*$, $\ell\in\mathbb R,$ while $[u]^t_{s,t}$ denotes the Gagliardo semi-norm given in (\ref{1.2}) below. $V_1(x), V_2(x), a(x): \R^N\to (0,\infty)$ are three positive weights, $M_1, M_2$ are continuous and positive functions in $\R^+$. Using variational methods, we prove existence of infinitely many high-energy solutions for the above system.
Abstract
In this paper, we investigate the fractional $p\&q$-Kirchhoff type system $$\left\{ \begin{array}{l} M_1([u]^p_{s,p})(-\Delta)_p^su+V_1(x)|u|^{p-2}u\\ \quad=\ell k^{-1}F_u(x,u,v)+\lambda a(x)|u|^{m-2}u,\quad x\in\Omega,\\ M_2([u]^q_{s,q})(-\Delta)_q^sv+V_2(x)|v|^{q-2}v\\ \quad=\ell k^{-1}F_v(x,u,v)+\mu a(x)|v|^{m-2}v,\quad\ x\in\Omega,\\ u\,=v\,=0,\qquad\qquad\qquad\qquad\qquad\qquad\ \ x\in\partial\Omega, \end{array} \right.$$ where $\Omega\subset \mathbb{R}^N$ is an unbounded domain with smooth boundary $\partial\Omega$, and $0<s<1<p\leq q$ and $sq<N,$ $\lambda,\mu>0$, $1<m\leq k<p_s^*$, $\ell\in\mathbb R,$ while $[u]^t_{s,t}$ denotes the Gagliardo semi-norm given in (\ref{1.2}) below. $V_1(x), V_2(x), a(x): \R^N\to (0,\infty)$ are three positive weights, $M_1, M_2$ are continuous and positive functions in $\R^+$. Using variational methods, we prove existence of infinitely many high-energy solutions for the above system.
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