Semilinear elliptic problems with nonlinear boundary conditions in unbounded domains (Q1909627)

It is considered the problem \[ - \Delta u+ a(x) u= g(x, u)\quad\text{in} \quad \Omega,\quad \partial_n u= \varphi(\xi, u)\quad \text{on} \quad \Gamma, \] where \(\Omega\) is an unbounded domain in \(\mathbb{R}^n\) \((n\geq 3)\) with smooth boundary \(\Gamma\), \(\partial_n\) denotes the outer normal derivative on \(\Gamma\), \(a\in L^\infty(\Omega)\) verifies \(a(x)\geq A> 0\) a.e. \(x\in \Omega\) and \(g\), \(\varphi\) have essentially the form \(|u|^{p- 1} u\) with \(p> 1\). The existence of a solution to this problem is obtained by a truncation approach and by using the mountain pass lemma.