Ψ-Hilfer fractional derivative as a generalization of many important nonlocal derivatives such as Riemann-Liouville, Caputo and Hadamard fractional derivatives, has a great importance in fractional calculations and theory of fractional differential equations. Accordingly, in this paper, we study the multiplicity results for Ψ-Hilfer fractional problems. Specially, our goal is to establish the existence of infinitely many nontrivial or distinct weak solutions for a nonlocal Ψ-Hilfer fractional problem by using critical point theory.

In this work, we study the multiplicity results for parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n (n\geq 2)$ and with zero Dirichlet boundary conditions. In fact, using a consequence of the local minimum theorem due to Bonanno we look into the existence one solution under algebraic conditions on the nonlinear term, and then by employing two critical point theorems, one due to due Averna and Bonanno, and another one due to Bonanno we guarantee the existence of two and three solutions for the problem $A_{1/2}u=\lambda f(u)$ in $\Omega$ with boundary condition $u=0$ on $\partial\Omega$.