The Moving Plane Method for Doubly Singular Elliptic Equations Involving a First-Order Term

In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first-order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov–Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.

Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

On the moving plane method for boundary blow-up solutions to semilinear elliptic equations

We consider weak solutions to {-\Delta u=f(u)} on {\Omega_{1}\setminus\Omega_{0}} , with {u=c\geq 0} in {\partial\Omega_{1}} and {u=+\infty} on {\partial\Omega_{0}} , and we prove monotonicity properties of the solutions via the moving plane method. We also prove the radial symmetry of the solutions in the case of annular domains.

A Note on Symmetry of Solutions for a Class of Singular Semilinear Elliptic Problems

We prove symmetry and monotonicity properties for positive solutions of the singular semilinear elliptic equationin bounded smooth domains with zero Dirichlet boundary conditions. The well-known moving plane method is applied.

Symmetry of solutions to optimization problems related to partial differential equations

We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.