Luigi Montoro

On the moving plane method for nonlocal problems in bounded domains

We consider a nonlocal problem involving the fractional Laplacian and the Hardy potential in bounded smooth domains. Exploiting the moving plane method and some weak and strong comparison principles, we deduce symmetry and monotonicity properties of positive solutions under zero Dirichlet boundary conditions.

Asymptotic symmetry for a class of quasi-linear parabolic problems

Abstract We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the ω limit set are nonnegative radially symmetric solutions of the stationary problem.

Symmetry results for the p(x)-Laplacian equation

Abstract. We consider the equation and the related Dirichlet problem. For axially symmetric domains we prove that, under suitable assumptions, there exist mountain-pass solutions which exhibit partial symmetry. Furthermore, we show that semi-stable or non-degenerate smooth solutions need to be radially symmetric in the ball.

Symmetry Results for Nonvariational Quasi-Linear Elliptic Systems

Abstract By virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.

Monotonicity and symmetry of singular solutions to quasilinear problems

We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved moving plane procedure.

Ireneo Peral: Forty Years as Mentor

Abstract In this article we present a survey of the Ph.D. theses that have been completed under the advice of Ireneo Peral. Following a chronological order, we summarize the main results contained in the works of the former students of Ireneo Peral.