Multiple positive solutions for a fractional $$ p \& q$$ -Laplacian system with concave and critical nonlinearities
Tóm tắt
In this paper, we study the following nonlinear fractional p &q-Laplacian system with critical exponent
$$\begin{aligned} {\left\{ \begin{array}{ll}(-\Delta )_p^{s_1} u+(-\Delta )_q^{s_2} u=\lambda \vert u\vert ^{r-2} u+\frac{2 \alpha }{\alpha +\beta }\vert u\vert ^{\alpha -2} u\vert v\vert ^{\beta }, &{} \text{ in } \Omega , \\ (-\Delta )_p^{s_1} v+(-\Delta )_q^{s_2} v=\mu \vert v\vert ^{r-2} v+\frac{2 \beta }{\alpha +\beta }\vert u\vert ^{\alpha }\vert v\vert ^{\beta -2} v, &{} \text{ in } \Omega , \\ u=v=0, &{} \text{ in } {\mathbb {R}}^{N} \backslash \Omega ,\end{array}\right. } \end{aligned}$$
where
$$\Omega $$
is a smooth bounded set in
$${\mathbb {R}}^{N}, \lambda , \mu >0$$
are two parameters,
$$0p s_{1}$$
,
$$\alpha ,\beta >1$$
satisfy
$$\alpha +\beta =p_{s_{1}}^{*}$$
with
$$p_{s_{1}}^{*}=\frac{n p}{n-p s_{1}}$$
is the fractional Sobolev critical exponent and
$$(-\Delta )_{t}^{s}$$
is the fractional t-Laplacian operator. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we show that the above system has at least
$$cat(\Omega )+1$$
distinct positive solutions, where
$$cat(\Omega )$$
denotes the Lusternik-Schnirelman category of
$$\Omega $$
in itself.
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