The author considers the elastic energy functional \(I[\overrightarrow{u}]=\int_{\Omega }\xi _{ij}(\overrightarrow{u})\xi_{ij}(\overrightarrow{u})\,dx\), where \(\Omega \) is a bounded domain in \(\mathbb{R}^{n}\), \(n>2\), with sufficiently smooth boundary \(\partial \Omega \) and \(\xi (\overrightarrow{u}) \) is the linearized strain tensor. The minimum of this functional is the free energy at equilibrium of a linear elastic material whose boundary displacements are zero. The author investigates the critical points of the functional \(J[\overrightarrow{u};\lambda ]=I[\overrightarrow{u}]-\lambda G[\overrightarrow{u}]\), where \(\lambda \) is a real parameter and \(G[ \overrightarrow{u}]=\int_{\partial \Omega }g(\overrightarrow{u},x)\,dS_{n-1}\) for some real function \(g\) defined on \(\partial \Omega \), which is supposed to be sufficiently differentiable and to satisfy \(g(\overrightarrow{0},x)=0\), \(\nabla _{\overrightarrow{u}}g(\overrightarrow{0},x)=0\), for all \(x\) on \(\partial \Omega \). The author first proves the existence of at least one minimizer \(\overrightarrow{u}\in \overset{\circ }{W}^{1,2}(\Omega\mathbb{R}^{n})\) of the energy functional \(I\), using Korn’s inequality. He also proves the existence of at least one minimizer of the energy functional \(I\) on the set \(\mathcal{B}=\{\overrightarrow{u}\in \overset{\circ }{W}^{1,2}(\Omega,\mathbb{R}^{n}):\left\vert \overrightarrow{u}(x)\right\vert =1\}\) and he derives the associated Euler-Lagrange equation. He defines a critical point of the functional \(J\) as a vector field \(\overrightarrow{u}\in W^{1,2}(\Omega ,\mathbb{R}^{n})\) such that \(J^{\prime }[\overrightarrow{u} ;\lambda ]\overrightarrow{v}=I^{\prime }[\overrightarrow{u}]\overrightarrow{v }-\lambda G^{\prime }[\overrightarrow{u}]\overrightarrow{v}\), for every \(\overrightarrow{v}\in W^{1,2}(\Omega ,\mathbb{R}^{n})\). The main result proves that \(\lambda _{0}\in \mathbb{R}\) is a bifurcation point for the preceding equation if and only if there exists \(\overrightarrow{u}_{0}\in W^{1,2}(\Omega ,\mathbb{R}^{n})\) with \(\overrightarrow{u}_{0}\neq 0\) such that \(\int_{\Omega }\xi _{ij}(\overrightarrow{u}_{0})\xi _{ij}(\overrightarrow{v})dx-\lambda _{0}\int_{\partial \Omega }g_{u^{j}}( \overrightarrow{0},x)u_{0}^{i}v^{j}dS_{n-1}=0\) for all \(\overrightarrow{v}\in W^{1,2}(\Omega ,\mathbb{R}^{n})\). In his final result, the author characterizes a bifurcation point of a functional associated to a decomposition of \(W^{1,2}(\Omega ,\mathbb{R}^{n})\) in a direct sum of two subspaces which involve the mean curvature of \(\partial \Omega \). He here uses a modified Skrypnik’s theory he exposed in [Int. J. Nonlinear Anal. Appl. 2, No. 1, 1–10 (2011; Zbl 1281.35003)].