Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

In this paper, we consider the second order discontinuous differential equation in the real line, ${\left(a\left(t,u\right)\varphi \left(u^{\prime }\right)\right)}^{\prime }=f\left(t,u,u^{\prime }\right),a.e.t\in \mathbb{R},u(-\infty )={\nu }^{-},u(+\infty )={\nu }^{+},$ with $\varphi$ an increasing homeomorphism such that $\varphi \left(0\right)=0$ and $\varphi \left(\mathbb{R}\right)=\mathbb{R}$ , $a\in C({\mathbb{R}}^2,\mathbb{R})$ with $a(t,x)>0$ for $(t,x)\in {\mathbb{R}}^2$ , $f:{\mathbb{R}}^3\to \mathbb{R}$ a $L^1$ -Carathéodory function and ${\nu }^{-},{\nu }^{+}\in \mathbb{R}$ such that ${\nu }^{-}<{\nu }^{+}$ . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities $\varphi$ and $f$ . To the best of our knowledge, this result is even new when $\varphi \left(y\right)=y$ , that is for equation ${\left(a\left(t,u\left(t\right)\right)u^{\prime }\left(t\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),a.e.t\in \mathbb{R}$ . Moreover, these results can be applied to classical and singular $\varphi$ -Laplacian equations and to the mean curvature operator. View Full-Text