Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions
We consider the elliptic problem with nonlinear boundary conditions: $$displaylines{ -Delta u +bu=f(x,u)quadhbox{in }Omega,cr -partial_{u}u=|u|^{q-1}u-g(u)quadhbox{on }partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^n$. Proving the existence of solutions of this problem relie...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2012-02-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2012/33/abstr.html |
| Summary: | We consider the elliptic problem with nonlinear boundary conditions: $$displaylines{ -Delta u +bu=f(x,u)quadhbox{in }Omega,cr -partial_{u}u=|u|^{q-1}u-g(u)quadhbox{on }partialOmega, }$$ 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 $L^{q+1}(partialOmega)subset H^1(Omega)$ 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. |
|---|---|
| ISSN: | 1072-6691 |