On the existence of three non-negative solutions for a $(p,q)$-Laplacian system

Abstract

The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-Δ)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-Δ)^{s_2}_{p_2} v &=\la f_2\,(x,u,v) +g_2(x,v) \,\mbox{ in }\, \Om, \\ u=v &= 0 \,\mbox{in }\, \Rn \setminus \Om, \\ \end{aligned} \right. \end{equation*} where $\Om \subset \Rn$ is a smooth bounded domain with smooth boundary, $s_1,s_2\in (0,1)$, $1<p_i<\frac{N}{s_i}$, $i=1,2$, $f_i$ and $g_i$ has certain growth assumptions for $i=1,2$. We prove existence of at least three non negative solutions of $(\mathcal{P})$ under restrictive range of $\lambda$ using variational methods. As a consequence, we also conclude that a similar result can be obtained when we consider a more general non local operator ${\mathcal{L}}_{{\varphi }_i}$ instead of $(-\Delta {)}_{p_i}^{s_i}$ in $(\mathcal{P})$.

References in corpus (1)

• Principal eigenvalue and positive solutions for Fractional $P-Q$ Laplace operator in quantum field theory