Nontrivial solutions for a fractional p-Laplacian problem via Rabier Theorem

By using a generalization of the Struwe–Jeanjean monotonicity trick we prove the existence of a non-trivial weak solution for the following problem \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s}_{p} u= f(x, u) &\mbox{ in } \Omega, \\ u=0 &\mbox{ in } \mathbb{R}^{N}\setminus \Omega \end{array} \right. \end{equation*} where $\Omega$ is a smooth bounded open set in $\mathbb{R}^{N}$, $p\in (1, \infty)$, $s\in (0, 1)$, $N>sp$, $(-\Delta)^{s}_{p}$ is the fractionsl $p$-Laplace operator and $f: \overline{\Omega}\times \mathbb{R}\times \mathbb{R}$ is a continuous function with $f(\cdot, 0)=0$ and there are $\alpha>p$ and $\sigma \in L^{1}(\Omega)$ such that $f(x, t)t \geq \alpha \int_{0}^{t} f(x, \tau) d\tau$ whenever $\int_{0}^{t} f(x, \tau) d\tau \geq \sigma(x)$.