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

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.