Second order elliptic equations of variational type in unbounded domains
Let \(\Omega\) be an unbounded domain of \(R^ n\), \(n\geq2\), endowed with the uniform \(C^2\)-regularity property. Let \(s\), \(t\), \(t_0\) be real numbers such that: \(s>2\) if \(n=2\), \(s=n\) if \(n>2\), \(t=2\) if \(2\leq n<4\), \(t>2\) if \(n=4\), \(t=n/2\) if \(n>4\), \(t_0 >t\). We consider in \(\Omega\) the linear second order uniformly elliptic operator: \[ Lu = -\sum^{n}_{i,j=1} a_{ij} u_{x_ i x_ j} + \sum^{n}_{i=1} a_ i u_{x_ i} + au. \] We suppose that: \[ \begin{gathered} a_{ij} = a_{ji}\in L^\infty(\Omega) \;cap W^{1,s}_{\text{loc}} (\bar\Omega), \quad i,j = 1,\ldots,n, \\ a_ i \in L^ s_{\text{loc}} (\bar \Omega), \quad \lim_{| x| \to +\infty} \| a_ i\|_{L^ s(\Omega \cap B(x,1))} = 0, \\ a\in L^{t_ 0}_{\text{loc}} (\bar \Omega), \quad \sup_{x\in\Omega} \| a\|_{L^{t_ 0}(\Omega \cap B(x,1))} < +\infty, \quad \inf_\Omega a > 0. \end{gathered} \] Moreover we suppose that the derivatives \((a_{ij})_{x_ k}\), \(i,j,k=1,\ldots,n\), have at the infinity the same behaviour of the \(a_ i\). In the above hypotheses, we prove that the Dirichlet problem: \[ u\in W^2 (\Omega) \cap \overset\circ W^1 (\Omega), \quad Lu=f, \quad f\in L^2 (\Omega), \] is uniquely solvable.
scientific article; zbMATH DE number 4103566
scientific article