# Existence and Multiplicity Results for an Elliptic Problem Involving Cylindrical Weights and a Homogeneous Term $$\mu$$

• R. B. Assunção
• O. H. Miyagaki
• L. C. Paes-Leme
• B. M. Rodrigues
Article

## Abstract

We consider the following elliptic problem
\begin{aligned} \left\{ \begin{array}{lll} -{\text {div}}\left( \dfrac{\left| \nabla u\right| ^{p-2} \nabla u}{\left| y\right| ^{ap}}\right) = \mu \dfrac{\left| u\right| ^{p-2} u}{\left| y\right| ^{p(a+1)}}+ h(x) \dfrac{\left| u\right| ^{q-2} u}{\left| y\right| ^{bq}} + f(x,u) &{}&{} \text{ in } \ \Omega , \\ u = 0 &{}&{} \text{ on } \ \partial \Omega ,\\ \end{array} \right. \end{aligned}
in an unbounded cylindrical domain
\begin{aligned} \Omega :=\{ (y,z)\in {\mathbb {R}}^{m+1}\times {\mathbb {R}}^{N-m-1} \ ; \ 0<A<\left| y\right|<B <\infty \}, \end{aligned}
where $$A,B\in {\mathbb {R}}_+$$, $$p>1$$, $$1\le m<N-p$$, $$q:=\dfrac{Np}{N-p (a+1-b)},$$ $$0\le \mu < {\overline{\mu }}:=\left( \dfrac{m+1-p(a+1)}{p}\right) ^p$$, $$h\in L^{\frac{N}{q}}(\Omega )\cap L^{\infty }(\Omega )$$ is a positive function and $$f: \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ is a Carathéodory function with growth at infinity. Using the Krasnoselski’s genus and applying $${\mathbb {Z}}_2$$ version of the Mountain Pass Theorem, we prove, under certain assumptions about f, that the above problem has infinite invariant solutions.

## Keywords

Supercritical degenerate operator variational methods

## Mathematics Subject Classification

35B07 35J62 35J70

## Notes

### Acknowledgements

O.H. Miyagaki received research grants from CNPq/Brazil Proc.304015/2014-8, FAPEMIG/Brazil CEX APQ-00063/15 and INCTMAT/CNPq/Brazil.

### Conflict of interest

The authors declare that they have no competing interests.

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## Authors and Affiliations

• R. B. Assunção
• 1
• O. H. Miyagaki
• 2
• L. C. Paes-Leme
• 3
• B. M. Rodrigues
• 3
Email author
1. 1.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil
2. 2.Departamento de MatemáticaUniversidade Federal de Juiz de ForaJuiz de ForaBrazil
3. 3.Departamento de MatemáticaUniversidade Federal de Ouro PretoOuro PretoBrazil