Existence and multiplicity results for a doubly anisotropic problem with sign-changing nonlinearity


We consider in this paper the following problem
 -\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,\\
 u=0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \text{on }\partial \Omega.
Where \Omega is a bounded regular domain in \mathbb{R}^{N}, % 1<p_{1}\leq p_{2}\leq ...\leq p_{N} and 1<q_{1}\leq q_{2}\leq ...\leq q_{N}, we will also assume that f is a continuous function, that have a finite number of zeroes, changing sign between them.

DOI Code: 10.1285/i15900932v39n2p1

Keywords: Anisotropic problem; exitence and mutiplicity; variational methods

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