Existence of nontrivial solutions for a fractional $p\&q$-Laplacian equation with sandwich-type and sign-changing nonlinearities

Abstract

In this paper, we deal with the following fractional $p\&q$ p & q -Laplacian problem: $$ \left \{ \textstyle\begin{array}{l@{\quad }l} (-\Delta )_{p}^{s}u +(-\Delta )_{q}^{s}u =\lambda a(x)|u|^{\theta -2}u+ \mu b(x)|u|^{r-2}u&\text{in}\;\ \Omega , \\ u(x)=0 &\text{in}\;\ \mathbb{R}^{N}\setminus \Omega , \end{array}\displaystyle \right . $$ { ( − Δ ) p s u + ( − Δ ) q s u = λ a ( x ) | u | θ − 2 u + μ b ( x ) | u | r − 2 u in Ω , u ( x ) = 0 in R N ∖ Ω , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N is a bounded domain with smooth boundary, $s\in (0,1)$ s ∈ ( 0 , 1 ) , $(-\Delta )_{m}^{s}$ ( − Δ ) m s $(m\in \{p,q\})$ ( m ∈ { p , q } ) is the fractional m-Laplacian operator, $p,q,r,\theta \in (1,p_{s}^{*}]$ p , q , r , θ ∈ ( 1 , p s ∗ ] , $p_{s}^{*}=\frac{Np}{N-sp}$ p s ∗ = N p N − s p , $\lambda , \mu >0$ λ , μ > 0 , and the weights $a(x)$ a ( x ) and $b(x)$ b ( x ) are possibly sign changing. Using the concentration compactness principle for fractional Sobolev spaces and the Ekeland variational principle, we prove that the problem admits a nonnegative solution for the critical case $r=p_{s}^{*}$ r = p s ∗ . Moreover, for the subcritical case $r< p_{s}^{*}$ r < p s ∗ , we obtain two existence results by applying the Ekeland variational principle and the mountain pass theorem.