Semilinear elliptic equations involving a gradient term in unbounded domains

Abstract

In this article, we study the existence of a classical solution of semilinear elliptic BVP involving gradient term of the type $$\displaylines{ -\Delta u=g(u)+\psi(\nabla u)+f\quad \text{ in }\Omega,\cr u=0\quad \text{on }\partial\Omega, }$$ where $\Omega$ is a (not necessarily bounded) domain in $\mathbb{R}^n$, $n\geq2$ with smooth boundary $\partial\Omega$. $f\in C_{\rm loc}^{0,\alpha}(\overline\Omega),0<\alpha<1$, $\psi\in C^{1}(\mathbb{R}^n,\mathbb{R})$ and $g$ satisfies certain conditions (well known in the literature as "jumping nonlinearity").

Keywords

• Monotone method
• Ambrosetti-prodi type problem
• unbounded domain