A perturbation theorem for the \(p\)-Laplace equation in unbounded domains (Q1612575)

This paper is devoted to the problem \[ \begin{cases} -\Delta_p u+ \lambda u^{p-1}= u^{q-1}\text{ in }\Omega\\ u>0\text{ in }\Omega,\;u\in W_0^{1, p} (\Omega),\end{cases} \tag{1} \] where \(\Omega\) is an unbounded domain in \(\mathbb{R}^N\), \(N\geq 3\), \(\Delta_p\) is the \(p\)-Laplacian, \(\lambda>0\), \(1<p<q <p^* ={N p\over N-p}\), \(W^{1,p}_0(\Omega)\) is the well-known Sobolev space. It is well-known that, if \(p=2\) and \(\Omega'\) is an axial-symmetric unbounded domain and \(\Omega\) is the perturbation of \(\Omega'\), moreover \(\Omega'\) and \(\Omega\) have the same topology, then there exists no solution of (1) in \(\Omega'\), but in \(\Omega\) (1) has a solution. Here, the authors extend this result to the case \(p>1\).