AbstractIn 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.
Numerical Enclosure-Methods for the Weak Solution of Elliptic Problems with Unilateral Constraints
