Multiplicity of critical points for some functionals related to the minimal surfaces problem (Q1384155)

The author establishes some existence and multiplicity results for critical points of the functional \(F_\varepsilon(u)=\varepsilon\int_\Omega|\nabla u|^2+\varepsilon^{-1}\int_\Omega G(u)\) (\(\varepsilon >0\), \(u\in H^1(\Omega)\)), where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^n\) and \(G\) is a positive \(C^2\) function having exactly two zeros, \(\alpha\) and \(\beta\). The main result of this paper asserts that if \(G\) satisfies a symmetry condition with respect to \((\alpha +\beta)/2\) then there exist two sequences of positive numbers \((\varepsilon_k)_k\) and \((c_k)_k\) such that for every \(\varepsilon\in(0,\varepsilon_k)\) the functional \(f_\varepsilon\) has at least \(k\) pairs of critical points \((u_{1,\varepsilon};\alpha+\beta-u_{1,\varepsilon}),\dots,(u_{k,\varepsilon};\alpha+\beta-u_{k,\varepsilon})\) which satisfy \(f_\varepsilon(u_i,\varepsilon)\leq c_k\), for every \(\varepsilon\in(0,\varepsilon_k)\) and any \(i=1,\dots,k\). In the case where \(G\) is not symmetric it is proved that for every \(\varepsilon>0\) at least one of the following two cases occurs: (i) there exist infinitely many minimizers, or (ii) \(f_\varepsilon\) has at least one saddle point.