In this article, we study the existence and uniqueness of distributional solution for semilinear fractional problems of Dirichlet form involving new operator. By means of the Leray–Schauder degree theory, we establish the existence results, with suitable assumptions on the semilinear term g and the contraction principle to prove the uniqueness in a particular case, then the numerical study of this problem using the finite difference method.

The main goal of this manuscript is to study the existence results in the Bessel Potential space for the following convection‐reaction fractional problem involving by distributional Riesz fractional derivative, {array−Dα.(Dαu(x))−divαbφ(u(x))=f(x,u(x),Dαu(x))arrayinΩ,arrayu=0arrayonℝd/Ω,$$ \left\{\begin{array}{cc}-{D}^{\alpha }.\left({D}^{\alpha }u(x)\right)- di{v}^{\alpha}\left( b\varphi \left(u(x)\right)\right)=f\left(x,u(x),{D}^{\alpha }u(x)\right)\kern0.60em & \kern0.1em \mathrm{in}\kern0.51em \Omega, \\ {}u=0\kern0.30em & \kern0.1em \mathrm{on}\kern0.51em {\mathbb{R}}^d/\Omega, \end{array}\right. $$ where normalΩ⊂ℝd$$ \Omega \subset {\mathbb{R}}^d $$ is a bounded open set with a Lipschitz boundary, α∈false(0,1false)$$ \alpha \in \left(0,1\right) $$ and d>2α$$ d>2\alpha $$. To reach our goal, we use the application of the Schauder fixed point theory with some assumption on the nonlinear terms φ$$ \varphi $$ and f$$ f $$.

In this paper, we investigate the existence of nontrivial solutions in the Bessel Potential space for nonlinearfractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using themountain pass theorem in combination with the perturbation method, we prove the existence of solutions.

In this paper, the study of the existence of a renormalized solution for quasilinear parabolicproblem with variable exponents and measure data. The model is: u_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda\left\vert u\right\vert ^{p(x)-2}u=\mu\text{ } &\text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[,\\u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\u(.,0)=u_{0}(.) & \text{in}\hspace{0.5cm}\Omega, where $ \lambda>0$ and $ T $ is any positive constant, $ \mu\in\mathcal{M}_{0}(Q) $ is any measure with bounded variation over $ Q=\Omega \times ]0,T[ $.