A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations
In this paper it is proved a critical point theorem of mountain-pass type which yields a sequence of critical points converging to zero. The proof combines a pseudo-gradient property with a deformation lemma. The abstract result is applied for proving a multiplicity result for the semilinear elliptic equation \(-\Delta u=f(x,u)\) in \(\Omega\) under the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with a smooth boundary and \(f(x,u)\) behaves like \(a(x)| u| ^p\)sgn\(\, u\) with \(0<p<1\).
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