Existence and multiplicity results for a doubly anisotropic problem with sign-changing nonlinearity
Abstract
We consider in this paper the following problem
Where
is a bounded regular domain in
,
and
we will also assume that
is a continuous function, that have a finite number of zeroes, changing sign between them.
![-\sum\limits_{i=1}^{N}\partial _{i}\left[ \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right] -\sum\limits_{i=1}^{N} \partial _{i}\left[ \left\vert \partial _{i}u\right\vert ^{q_{i}-2}\partial _{i}u\right] =\lambda f(u) \text{ \ }\ \text{in } \Omega,\\](http://siba-ese.unile.it/plugins/generic/latexRender/cache/20e2262caa4b1f4c64af3b33c86c6b57.png)
![u=0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \text{on }\partial \Omega.](http://siba-ese.unile.it/plugins/generic/latexRender/cache/efd19dda29740e7ae99a3afe68dba326.png)
Where
![\Omega](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b9d99db8626de63193c7fe96273a6cae.png)
![\mathbb{R}^{N}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/5cc1d5e761386d5fe710282ea8108641.png)
![% 1<p_{1}\leq p_{2}\leq ...\leq p_{N}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f863514073003a5c2d297786bfdad912.png)
![1<q_{1}\leq q_{2}\leq ...\leq q_{N},](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a1ff6147b2ad9b2df049ff8c959fc9c6.png)
![f](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8fa14cdd754f91cc6554c9e71929cce7.png)
DOI Code:
10.1285/i15900932v39n2p1
Keywords:
Anisotropic problem; exitence and mutiplicity; variational methods
Full Text: PDF