In this work we consider the fractional Kirchhoff equations with singular nonlinearity, $$\displaylines{ M\Big( \int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dx dy\Big) (-\Delta)^s_p u = \lambda a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\cr \int_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^p}{|x-y|^{N+sp}}dx v \mu b(x)|v|^{q-2}v +\fra...

In this paper, we first establish a narrow region principle for systems involving the fractional Laplacian in unbounded domains, which plays an important role carrying on direct method of moving planes. Then combining with sliding method, derive monotonicity bounded positive solutions to following Lipschitz domains \begin{document}$...

We consider the \(p\)-Laplacian system $$ \displaylines{ -\Delta_p u = \lambda f(v) \quad \text{in } \Omega; \cr v g(u) v=0 \text{on }\partial \Omega, }$$ where \(\lambda >0\) is a parameter, \(\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)\) operator for \(p > 1\) and \(\Omega\) unit ball in \(\mathbb{R}^N\) (\(N \geq 2)\). The nonlinearities \(f, g: [0,\infty) \to \mathbb{R}\...

Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p–Laplacian.

Recently, a method based on Laplacian eigenfunctions was proposed to automatically construct a basis for value function approximation in MDPs. We show that its success may be explained by drawing a connection between the spectrum of the Laplacian and the value function of the MDP. This explanation helps us to identify more precisely the conditions that this method requires to achieve good perfo...