Here, the anisotropic $ (p, q) $-Laplacian$ - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right) - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{q_i-2}\frac{\partial u}{\partial x_i}\right) = \lambda u^{\gamma-1} $is considered, where $ \Omega $ is a bounded and regular domain of $ \mathbb{R}^N $, $ q_i\leq p_i $ for $ i = 1, \cdots, N $ and $ \gamma > 1 $. The existence of positive solution is proved via sub-supersolution method.

Project Euclid - mathematics and statistics online

The existence of subsonic detonation wave (processes occurring at high speeds) for a system that is related to the combustion model for highly hydraulically resistant porous media is studied. The porous media combustion model consists of a system of coupled partial differential equations. We apply some topological arguments to prove the existence of heteroclinic orbits for the system which governs ordinary differential equations.

The existence of at least one positive radial solution of the Neumann problem $$ -\Delta _{\mathbb{H}^{n}} u+R(\xi ) u=a \bigl( \vert \xi \vert _{\mathbb{H}^{n}} \bigr) \vert u \vert ^{p-2} u - b\bigl( \vert \xi \vert _{\mathbb{H}^{n}}\bigr) \vert u \vert ^{q-2}u, $$−ΔHnu+R(ξ)u=a(|ξ|Hn)|u|p−2u−b(|ξ|Hn)|u|q−2u, is proved on the Heisenberg group $\mathbb{H}^{n}$Hn , via the variational principle, where $a(|\xi |_{\mathbb{H}^{n}})$a(|ξ|Hn) , $b(|\xi |_{\mathbb{H}^{n}})$b(|ξ|Hn) are nonnegative radial functions and $R(\xi )$R(ξ) satisfies suitable conditions.

In this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces.