Electron. J. Diff. Eqns., Vol. 2009(2009), No. 43, pp. 1-13.

Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions

Bhatia Sumit Kaur, K. Sreenadh

Abstract:
Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with $C^2$ boundary. In this paper, we are interested in the problem
$$\displaylines{ -\Delta u+u = h(x,u) e^{u^2}/|x|^\beta,\quad u>0 \quad \hbox{in } \Omega, \cr \frac{\partial u}{\partial\nu}= \lambda \psi u^q \quad \hbox{on }\partial \Omega, }$$
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.

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  Bhatia Sumit Kaur
Department of Mathematics
Indian Institute of Technology Delhi Hauz Khaz
New Delhi-16, India
email: sumit2212@gmail.com
Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi Hauz Khaz
New Delhi-16, India
email: sreenadh@gmail.com

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