Electron. J. Diff. Equ., Vol. 2012 (2012), No. 33, pp. 1-9.

Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions

Junichi Harada, Mitsuharu Otani

Abstract:
We consider the elliptic problem with nonlinear boundary conditions:
$$\displaylines{ -\Delta u +bu=f(x,u)\quad\hbox{in }\Omega,\cr -\partial_{\nu}u=|u|^{q-1}u-g(u)\quad\hbox{on }\partial\Omega, }$$
where $\Omega$ is a bounded domain in $\mathbb{R}^n$. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since $L^{q+1}(\partial\Omega)\subset H^1(\Omega)$ does not hold for large q, the standard variational method can not be applied directly. To overcome this difficulty, we use approximation methods and uniform a priori estimates for solutions of approximate equations.

Submitted November 17, 2011. Published February 23, 2012.
Math Subject Classifications: 35J20.
Key Words: Nonlinear boundary conditions

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  Junichi Harada
Department of Applied Physics
School of Science and Engineering
Waseda University, 3-4-1 Okubo
Shinjuku-ku Tokyo, 169-8555, Japan
email: harada-j@aoni.waseda.jp
Mitsuharu Ôtani
Department of Applied Physics
School of Science and Engineering
Waseda University, 3-4-1 Okubo
Shinjuku-ku Tokyo, 169-8555, Japan
email: otani@waseda.jp

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