Partial symmetry and existence of least energy solutions to some nonlinear elliptic equations on Riemannian models (Q745900)

On a suitable Riemannian manifold \((M,g)\) equipped with an almost explicit Riemannian metric, the authors study solution properties for \[ (*)\qquad -\Delta_gu=f(r,u) \] such that \(f\) is a continuous derivable function on \([0,\infty)\times \mathbb R\) and satisfying some conditions. Precisely, if \(u(r,\sigma)\) is a solution for \((*)\), then it is either symmetric with respect to \(r\in (0,R)\) (\(R\) can be a real number or infinity) or definite strictly monotone with respect to \(\sigma:=\arccos(\theta.\theta_0)\in [0,\pi]\) where \(\theta.\theta_0\) is the scalar product of \(\theta_0\) a fixed point on the unit sphere and \(\theta\in \mathbb S^{n-1}\), see Theorems 3.2 and 3.4, and their proofs are based on a few lemmas, e.g., in regard to the definiteness, see Lemma 4.1. Also, they state the related open problem. In Subsection 3.2, the authors deal with the existence of critical points of the action functional associated to \((*)\). Therefore, in a suitable complete Riemannian manifold endowed with bounded geometry, i.e., the injectivity radius is strictly positive and each covariant derivative of the curvature tensor is bounded, they show that \[ (**)\qquad -\Delta_gu=V(x)u+h(u) \] has a critical point whenever \(V\) is lower bounded by minus infinity and upper bounded by the first eigenvalue associated to \(-\Delta_g\) and \(h\) is a continuous derivable function on \(\mathbb R\) and satisfies some conditions (Proposition 3.7). Then they state another result that when a weakly homogeneous complete Riemannian manifold (Definition 3.6) is either a hyperbolic manifold or a Euclidean manifold (Proposition 3.8). Next, by imposing conditions on \(f\), the authors state that \((*)\) has a critical point, see Theorem 3.9, and the proof is based on the use both the Palais-Smale properties (Lemma 5.5) and the Ekeland variational principle. Proposition 3.12 states sufficient conditions for not having nontrivial solution for \((*)\).