In the present paper we de…ne a q-analogue of the modi…ed Bernstein-Kantorovich operators introduced by Özarslan and Duman(Numer. Funct. Anal. Optim. 37:92-105,2016). We establish the shape preserving properties of these operators e.g. monotonicity and convexity and study the rate of convergence by means of Lipschitz class and Peetre's K-functional and degree of approximation with the aid of a smoothing process e.g Steklov mean. Further, we introduce the bivariate case of modi…ed q-Bernstein-Kantorovich operators and study the degree of approximation in terms of the partial and total modulus of continuity and Peetre's K-functional. Finally, we introduce the associated GBS (Generalized Boolean Sum) operators and investigate the approximation of the Bögel continuous and Bögel di¤erentiable functions by using the mixed modulus of smoothness and Lipschitz class.

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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.

We establish Lyapunov‐type inequalities for a class of singular elliptic partial differential equations. As an application of Lyapunov‐type inequalities, we obtain lower bounds for the first eigenvalue of the associated singular elliptic partial differential equations. We also make an attempt to answer a question raised by de Nápoli and Pinasco in J. Funct. Anal. 270 (2016), no. 6, 1995–2018.

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