The Nehari Manifold for a Quasilinear Elliptic Equation with Singular Weights and Nonlinear Boundary Conditions
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Authors
S. H. Rasouli
- Department of Mathematics, Faculty of Basic Science, Babol University of Technology, Babol, Iran
K. Fallah
- Department of Mathematics, Islamic Azad University Ghaemshahr branch, Iran
Abstract
Using the technique of Brown and Wu [11]; we present a note on the paper [22] by Wu.
Indeed, we extend the multiplicity results for a class of semilinear problems to the quasilinear
elliptic problems with singular weights of the form:
\[
\begin{cases}
-div(|x|^{-ap}|\nabla u|^{p-2}\nabla u)\lambda|x|^{-(a+1)p+c}f(x)|u|^{q-2}u,\,\,\,\,\, x\in \Omega,\\
|\nabla u|^{p-2} \frac{\partial u}{\partial n}=|x|^{-(a+1)p+c}g(x)|u|^{r-2}u,
\,\,\,\,\, x\in \partial \Omega.
\end{cases}
\]
Here \(0\leq a<\frac{N-p}{p}, c\) is a positive parameter, \(1 < q < p < r < p*(p* = \frac{pN}{N-p}\) if \(N > p,
p* =\infty\) if \(N \leq p), \Omega\subset R^N\) is a bounded domain with smooth boundary, \(\frac{\partial }{\partial n}\) is the outer
normal derivative, \(\lambda\in R-{0}\); and \(f(x); g(x)\) are continuous functions which change sign
in \(\overline{\Omega}\).
Share and Cite
ISRP Style
S. H. Rasouli, K. Fallah, The Nehari Manifold for a Quasilinear Elliptic Equation with Singular Weights and Nonlinear Boundary Conditions, Journal of Mathematics and Computer Science, 3 (2011), no. 2, 262--277
AMA Style
Rasouli S. H., Fallah K., The Nehari Manifold for a Quasilinear Elliptic Equation with Singular Weights and Nonlinear Boundary Conditions. J Math Comput SCI-JM. (2011); 3(2):262--277
Chicago/Turabian Style
Rasouli, S. H., Fallah, K.. "The Nehari Manifold for a Quasilinear Elliptic Equation with Singular Weights and Nonlinear Boundary Conditions." Journal of Mathematics and Computer Science, 3, no. 2 (2011): 262--277
Keywords
- Quasilinear elliptic problem
- Singular weights
- Nehari manifold
- Nonlinear boundary condition.
MSC
- 35J62
- 35J20
- 35J50
- 35J30
- 35J48
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