Existence, multiplicity and regularity of solutions for the fractional p-Laplacian equation

J. Korean Math. Soc. Published online February 27, 2020

Yun-Ho Kim
Sangmyung University

Abstract : We are concerned with the following elliptic equations:
\begin{equation*}
\begin{cases}
(-\Delta)_p^su=\lambda f(x,u) \quad \text{in} \ \
\Omega,\\
u= 0\quad \text{on}\ \ \mathbb{R}^N\backslash\Omega,
\end{cases}
\end{equation*}
where $\lambda$ is real parameter, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0<s<1<p<+\infty$, $sp<N$, and $f:\Omega\times\Bbb R \to \Bbb R$ satisfies a Carath\'eodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters $\lambda$ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity $f$ has the subcritical growth condition. In addition, under adequate conditions, we establish an a-priori estimate in $L^{\infty}(\Omega)$ of any possible weak solution by applying the bootstrap argument.