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We establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations driven by arbitrary cylindrical L\'evy processes in Hilbert spaces. The coefficients are assumed to satisfy global…

In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We…

We prove existence and uniqueness of strong (pointwise) solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of Euclidean space. The system of equations comes from a linearization of a nonlocal model…

We develop a theory of generalized solutions of the nonlinear evolution equations for complex-valued functions of a real positive time variable and $p$-adic spatial variable, which can be seen as non-Archimedean counterparts of the…

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional $p-$Laplacian operator. Just…

In this paper, we investigate the existence and uniqueness of a non-trivial solution for a class of nonlocal equations involving the fractional $p$-Laplacian operator defined on compact Riemannian manifold, namely,…

We study a superlinear elliptic boundary value problem involving the $p$-laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the…

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}<p<2$.…

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional $p$-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration…

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation $$ (-\Delta)^s_p u=\lambda |u|^{p-2}u + f(x,u,\lambda) \quad\text{in}\quad \Omega,\quad u=0 \quad\text{in}\quad \mathbb{R}^n\setminus\Omega, $$…

We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We…

We consider a class of elliptic and parabolic problems, featuring a specific nonlocal operator of fractional-laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely…

In this article we implement a method for the computation of a nonlinear elliptic problem with nonstandard growth driven by the $p(x)-$Laplacian operator. Our implementation is based in the {\em decomposition--coordination} method that…

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal…

Since the pioneering works by Aronson & B\'enilan [C. R. Acad. Sci. Paris S\'er., 1979] and B\'enilan & Crandall [Johns Hopkins Univ. Press, 1981], it is well-known that first-order evolution problems governed by a nonlinear but homogeneous…

We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some…

We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in \cite{bcr}. Our main assumption is an appropriate…

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…

We study a pseudo-differential equation driven by the degenerate fractional p-Laplacian, under Dirichlet type conditions in a smooth domain. First we show that the solution set within the order interval given by a sub-supersolution pair is…