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 Solution to Singular Elliptic Problems with Subcritical nonlinearities

In this paper, we study the existence of a non-trivial weak solution to the following singular elliptic equations with subcritical nonlinearities:\left\{ {\matrix{ { - div\left( {{{\left| x \right|}^{ - 2\beta }}\nabla u} \right) - \mu {{f(x)u} \over {{{\left| x \right|}^{2(\beta + 1)}}}} = {{\lambda g(x)} \over {{u^\theta }}} + h(x){u^p}\,\,\,\,in\,\,\,\Omega ,} \hfill \cr {u > 0\,\,\,in\,\,\Omega ,} \hfill \cr {u = 0\,\,on\,\,\partial \Omega ,} \hfill \cr } } \right.where Ω ⊂ℝN is an open bounded domain with C1 boundary, θ, λ > 0, 0 < \beta < {{N - 2} \over 2} 0< p< 1, 0 < \mu < {\left( {{{N - 2(\beta + 1)} \over 2}} \right)^2}, N ≥ 3, 0 ∈ Ω and 0 ≤ f, g, h ∈ L∞ (Ω). We show that there exists a solution u \in H_0^1\left( {\Omega ,{{\left| x \right|}^{ - 2\beta }}} \right) \cap {L^\infty }(\Omega ) to this problem.