We deal with Dirichlet problems of the form {Δ⁢u+f⁢(u)=0in ⁢Ω,u=0on ⁢∂⁡Ω,\left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝn{\mathbb{R}^{n}}, n≥3{n\geq 3}, and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain Tε⁢(Γk){T_{\varepsilon}(\Gamma_{k})} with thickness ε>0{{\varepsilon}>0} and center Γk{\Gamma_{k}}, a k -dimensional, smooth, compact submanifold of ℝn{\mathbb{R}^{n}}. Our main result concerns the case where k=1{k=1} and Γk{\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε>0{{\varepsilon}>0} small enough. When k≥2{k\geq 2} or Γk{\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f .

中文翻译：

---

管域中超临界增长Dirichlet问题解的不存在

我们处理形式为{Δproblemsu +f⁢（u）= 0 inΩ，u = 0 onΩ，\ left \ {\ begin {aligned} \ displaystyle {} \ Delta u + f（ u）＆\ displaystyle = 0 &&％\ displaystyle \ phantom {} \ text {in} \ Omega，\\ \ displaystyle u＆\ displaystyle = 0 && \ displaystyle \ phantom {} \ text {on} \ partial％\ Omega，\ end {aligned} \ right。其中，Ω是isn {\ mathbb {R} ^ {n}}的有界域，n≥3{n \ geq 3}，从Sobolev嵌入的角度来看，f具有超临界增长。特别地，我们考虑Ω是管状区域Tε⁢（Γk）{T _ {\ varepsilon}（\ Gamma_ {k}）}，厚度为ε> 0 {{\ varepsilon}> 0}，中心Γk{\ Gamma_ {k}}，{n {\ mathbb {R} ^ {n}}的ak维，光滑，紧凑的子流形。我们的主要结果涉及k = 1 {k = 1}和Γk{\ Gamma_ {k}}本身可收缩的情况。在这种情况下，我们证明对于ε> 0 {{\ varepsilon}> 0}足够小的问题，该问题没有非平凡的解。当k≥2{k \ geq 2}或Γk{\ Gamma_ {k}}本身是不可约的时，我们得到的弱的不存在结果。一些示例表明，对于涉及k和f的假设，所有这些结果都是清晰的。