| Summary: | Abstract We consider the following elliptic problem: { − div ( | ∇ u | p − 2 ∇ u | y | a p ) = | u | q − 2 u | y | b q + f ( x ) in Ω , u = 0 on ∂ Ω , $$ \textstyle\begin{cases} -\operatorname{div} ( \frac{ \vert \nabla u \vert ^{p-2} \nabla u}{ \vert y \vert ^{ap}} ) = \frac { \vert u \vert ^{q-2} u}{ \vert y \vert ^{bq}} + f(x) & \mbox{in } \Omega,\\ u = 0 & \mbox{on } \partial\Omega, \end{cases} $$ in an unbounded cylindrical domain Ω : = { ( y , z ) ∈ R m + 1 × R N − m − 1 ; 0 < A < | y | < B < ∞ } , $$\Omega:= \bigl\{ (y,z)\in \mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1} ; 0< A< \vert y \vert < B < \infty \bigr\} , $$ where 1 ≤ m < N − p $1\leq m< N-p$ , q = q ( a , b ) : = N p N − p ( a + 1 − b ) $q=q(a,b):=\frac{Np}{N-p(a+1-b)}$ , p > 1 $p>1$ and A , B ∈ R + $A,B\in\mathbb{R}_{+}$ . Let p N , m ∗ : = p ( N − m ) N − m − p $p^{*}_{N,m}:=\frac {p(N-m)}{N-m-p}$ . We show that p N , m ∗ $p^{*}_{N,m}$ is the true critical exponent for this problem. The starting point for a variational approach to this problem is the known Maz’ja’s inequality (Sobolev Spaces, 1980) which guarantees, for the q previously defined, that the energy functional associated with this problem is well defined. This inequality generalizes the inequalities of Sobolev ( p = 2 , a = 0 and b = 0 ) $(p=2, a=0 \mbox{ and } b=0)$ and Hardy ( p = 2 , a = 0 and b = 1 ) $(p=2, a=0 \mbox{ and } b=1)$ . Under certain conditions on the parameters a and b, using the principle of symmetric criticality and variational methods, we prove that the problem has at least one solution in the case f ≡ 0 $f\equiv0$ and at least two solutions in the case f ≢ 0 $f \not\equiv0$ , if p < q < p N , m ∗ $p< q< p^{*}_{N,m}$ .
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