The author proves existence results for solutions of the equation \( A_{1/2}w=\mu w+b(x)| w| ^{\gamma -1}w\) posed in a smooth and bounded domain \(\Omega \) of \(\mathbb{R}^{N}\), \(N\geq 2\), with homogeneous Dirichlet boundary conditions on \(\partial \Omega \). Here \( A_{1/2}\) denotes the square root of the Laplace operator in \(\Omega \) with homogeneous Dirichlet boundary conditions on \(\partial \Omega \). This operator \(A_{1/2}\) is defined through \(A_{1/2}u=\sum_{k=1}^{\infty }a_{k}\lambda _{k}^{1/2}\varphi _{k}\) where \((\lambda _{k},\varphi _{k})_{k}\) is the sequence of eigenelements of the Laplace operator \(-\Delta \) and \( u=\sum_{k=1}^{\infty }a_{k}\varphi _{k}\in L^{2}(\Omega )\). In the above equation \(\mu \) and \(\gamma \) are parameters and \(b(x)\) is a sign-changing continuous function on \(\Omega \). The author first links the above equation to the boundary value problem (BVP) \(-\Delta u=0\) in \(\mathcal{C}=\Omega \times (0,\infty )\) with \(u=0\) on \(\partial _{L}\mathcal{C}=\partial \Omega \times \left[ 0,\infty \right) \) and \(\frac{\partial u}{\partial \nu }=\mu u+b(x)| u| ^{\gamma -1}u\) on \(\Omega \times \{0\}\). The solutions of the BVP can be seen as critical points of the functional \( I_{\mu }\) from \(H=H_{0,L}^{1}(\mathcal{C})=\{v\in H^{1}(\mathcal{C})\mid v=0\) on \(\partial \mathcal{C}\}\) to \(\mathbb{R}\) and defined through \[ I_{\mu }(u)= \frac{1}{2}\int_{\mathcal{C}}| \nabla u| ^{2}dx-\frac{\mu }{2}\int_{\Omega \times \{0\}}| u| ^{2}dx-\frac{1}{\gamma +1}\int_{\Omega \times \{0\}}b(x)| u(x,0)| ^{\gamma +1}dx. \] The author introduces the Nehari manifold \(\mathcal{S}\) associated to \( I_{\mu }\) as \(S=\{u\in H\mid \int_{\mathcal{C}}| \nabla u| ^{2}dx-\int_{\Omega \times \{0\}}| u| ^{2}dx=\int_{\Omega \times \{0\}}b(x)| u(x,0)| ^{\gamma +1}dx\}\). The author first considers the superlinear case assuming \( 10\), which is a minimizer of \(I_{\mu }\) on the subset \(\mathcal{S}^{-}\) of \( \mathcal{S}\) corresponding to the set of local maxima. The behaviour of this solution and of its energy when \(\mu \to \mu _{1}^{-}\) is described. When \(\int_{\Omega \times \{0\}}b(x)| \varphi _{1}(x,0)| ^{\gamma +1}dx0\). The behaviour of the minimizer \(u_{n}\) of \(I_{\mu _{n}}\) on the subset \( \mathcal{S}^{+}\) of local minima of \(\mathcal{S}\) when \(\mu _{n}\to \mu _{1}^{+}\). Finally, for this superlinear case, the author observes that when \(\int_{\Omega \times \{0\}}b(x)| \varphi _{1}(x,0)| ^{\gamma +1}dx>0\) the quantity \(\inf_{u\in \mathcal{S}^{-}}I_{\mu }(u)=0\) for all \(\mu \to \mu _{1}\). In the sublinear case, \(00\) and \(\int_{\Omega \times \{0\}}b(x)| \varphi _{1}(x,0)| ^{\gamma +1}dx<0\). The author first establishes some properties of \(\mathcal{S}^{+}\) and \(\mathcal{S}^{-}\). The proof of the existence result is mainly obtained studying the behaviour of minimizing sequences.