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

We are concerned with the following elliptic equations: { (−∆)pu = λf(x, u) in Ω, u = 0 on RN\Ω, where λ are real parameters, (−∆)p is the fractional p-Laplacian operator, 0 < s < 1 < p < +∞, sp < N , and f : Ω × R → R satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ 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 apriori estimate in L∞(Ω) of any possible weak solution by applying the bootstrap argument.