Electron. J. Diff. Equ., Vol. 2014 (2014), No. 219, pp. 1-13.

Semilinear elliptic equations involving a gradient term in unbounded domains

V. Raghavendra, Rasmita Kar

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").

Submitted October 10, 2011. Published October 16, 2014.
Math Subject Classifications: 35J65, 35J25.
Key Words: Monotone method; Ambrosetti-prodi type problem; unbounded domain.

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Venkataramanarao Raghavendra
Department of Mathematics and Statistics
Indian Institute of Technology
Kanpur 208016, India
email: vrag@iitk.ac.in
Rasmita Kar
Department of Mathematics and Statistics
Indian Institute of Technology
Kanpur 208016, India
email: rasmitak6@gmail.com

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