Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 43, pp. 1-13.
Title: Multiple positive solutions for a singular elliptic equation
with Neumann boundary condition in two dimensions
Authors: Bhatia Sumit Kaur (Indian Inst. of Technology Delhi, India)
K. Sreenadh (Indian Inst. of Technology Delhi, India)
Abstract:
Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with $C^2$
boundary. In this paper, we are interested in the problem
$$\displaylines{
-\Delta u+u = h(x,u) e^{u^2}/|x|^\beta,\quad
u>0 \quad \text{in } \Omega, \cr
\frac{\partial u}{\partial\nu}= \lambda \psi u^q \quad
\text{on }\partial \Omega,
}$$
where $0\in \partial \Omega$, $\beta\in [0,2)$, $\lambda>0$, $q\in [0,1)$
and $\psi\ge 0$ is a H\"older continuous function on
$\overline{\Omega}$.
Here $h(x,u)$ is a $C^{1}(\overline{\Omega}\times \mathbb{R})$ having
superlinear growth at infinity. Using variational methods we show
that there exists $0<\Lambda <\infty$ such that above problem
admits at least two solutions in $H^1(\Omega)$ if
$\lambda\in (0,\Lambda)$, no solution if $\lambda>\Lambda$ and at least
one solution when $\lambda = \Lambda$.
Submitted August 26, 2008. Published March 24, 2009.
Math Subject Classifications: 34B15, 35J60
Key Words: Multiplicity; nonlinear Neumann boundary condition;
Laplace equation.