Krasnoselskii’s Genus in Variational Analysis

• Krasnoselskii’s genus is a topological invariant that measures the symmetric complexity of closed, nonzero subsets in Banach spaces.
• It is used to construct minimax levels and establish Palais–Smale sequences, linking symmetry properties to the multiplicity of critical points.
• The concept is crucial in nonlinear analysis, especially for even functionals in elliptic PDEs and nonsmooth settings.

Krasnoselskii's genus is a fundamental topological invariant for closed, symmetric subsets of a Banach (or more generally, topological vector) space, playing a central role in variational methods for proving the existence and multiplicity of critical points of even functionals. Genus theory codifies the idea of "symmetric complexity" and is particularly suited for problems exhibiting an inherent symmetry, especially problems involving functionals that are even (i.e., invariant under $u \mapsto -u$). It is extensively employed in nonlinear analysis, critical point theory, and the study of elliptic partial differential equations, especially in the context of finding infinitely many solutions.

1. Definition and Fundamental Properties

For a Banach space $X$ and $\mathcal{A}$ the class of closed, symmetric (i.e., $A = -A$), nonempty subsets $A \subset X \setminus \{0\}$, the Krasnoselskii genus $\gamma(A)$ is the least integer $k$ such that there exists an odd continuous map $\varphi: A \to \mathbb{R}^k \setminus \{0\}$. If such $k$ does not exist, set $\gamma(A) = \infty$, and by convention $\gamma(\emptyset) = 0$.

The main properties of the genus include:

• Monotonicity: If $A \subset B$, then $\gamma(A) \leq \gamma(B)$.
• Subadditivity: $\gamma(A \cup B) \leq \gamma(A) + \gamma(B)$.
• Genus of spheres: $\gamma(S^{n-1}) = n$ (where $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$).
• Infinite cardinality for genus $\geq 2$: If $\gamma(A) \geq 2$, then $A$ is infinite; thus, genus at least two forces an infinite set [(Figueiredo et al., 2015), Section 4].

These properties enable systematic quantification of symmetric subsets' topology, facilitating the construction of minimax critical levels and multiplicity results for variational systems.

2. Application in Variational Settings

Variational problems with even energy functionals benefit directly from genus theory. In "On a nonlocal multivalued problem in an Orlicz–Sobolev space via Krasnoselskii’s genus," genus-based arguments are used to analyze the multiplicity of solutions for a nonlocal multivalued elliptic inclusion in the Orlicz–Sobolev space $W_0^1L_\Phi(\Omega)$, where $\Phi$ is an $N$-function and the involved functionals are even and locally Lipschitz but not $C^1$.

Given a locally Lipschitz, even, coercive functional $J: X \to \mathbb{R}$ (in this context, $J$ is defined as

$$J(u) = \widehat{M}\left(\int_\Omega \Phi(|\nabla u|)\right) - \int_\Omega \Phi(u) - \int_\Omega F(u),$$

where $\widehat{M}(s) = \int_0^s M(t)dt$ and $F(t) = \int_0^t f(s)ds$), the genus is instrumental in constructing minimax levels associated with symmetric sets of high genus, leading to the extraction of distinct critical values [(Figueiredo et al., 2015), §§3–4].

3. Genus-Level Minimax Construction

For each $k \in \mathbb{N}$, define

$$\Gamma_k = \{C \subset X: C \text{ closed},\ C = -C,\, \gamma(C) \geq k\}$$

and set the $k$-th minimax level as

$c_k = \inf_{C \in \Gamma_k} \sup_{u \in C} J(u).$

This construction yields:

• Each $c_k$ is finite and, under the functional's coercivity and other structural conditions, negative for all $k \geq 1$.
• For each $k$, $J$ admits a Palais–Smale sequence at level $c_k$, and the set of critical points at $c_k$, denoted $K_{c_k}$, is nonempty, compact, and symmetric.

Genus theory enables the following dichotomy (cf. Lemma 4.5, Lemma 4.7, and Lemma 4.9 in (Figueiredo et al., 2015)):

• If the sequence $(c_k)$ is strictly increasing, there are infinitely many distinct critical levels, yielding infinite multiplicity.
• If there exists $k$ with $c_k = c_{k+1} = \cdots = c_{k+r}$, then $\gamma(K_{c_k}) \geq r+1 \geq 2$, which, by the genus properties, forces $K_{c_k}$ to be infinite; again, there are infinitely many solutions.

4. Critical Point Theory in Nonsmooth Contexts

Krasnoselskii's genus is amenable to the analysis of functionals lacking $C^1$ regularity. In the outlined nonlocal multivalued problem, $J$ is only locally Lipschitz; thus, critical points are defined via Clarke's subdifferential: $0 \in \partial J(u)$. The genus is nonetheless fully applicable, as long as Palais–Smale type compactness holds. Lemma 4.2 in (Figueiredo et al., 2015) establishes that $J$ satisfies the nonsmooth (PS) condition, ensuring the existence of strongly convergent Palais–Smale sequences and facilitating the genus-based minimax arguments even in the absence of Fréchet differentiability.

5. Consequences for Multiplicity of Solutions

The genus-based method systematically yields infinitely many nontrivial solutions for variational problems governed by even, locally Lipschitz energy functionals. In the Orlicz–Sobolev setting considered in (Figueiredo et al., 2015), Theorem 1.1 demonstrates that, under suitable hypotheses on the nonlinearity and the nonlocal term, as well as a smallness condition on the parameter $a_0$, the nonlocal multivalued elliptic problem has infinitely many solutions in $W_0^1L_\Phi(\Omega)$. For each $k$, there are at least $k$ different critical points at levels $c_1, \ldots, c_k$, establishing a direct link between genus and solution multiplicity.

Moreover, each critical point is a genuine solution: with $a_0$ small enough, the measure $\{x : u(x) > a_0\}$ is positive (cf. condition (4.17) in the paper), validating nontriviality.

6. Structural Role in Nonlinear Analysis

Krasnoselskii's genus serves as an essential instrument in nonlinear analysis for quantifying symmetric topological complexity and constructing multiplicity results via minimax and critical point theory. Its robust behavior under continuous mappings and set-theoretic operations, along with compatibility with nonsmooth analysis and compactness principles, underpins modern variational approaches to elliptic PDEs—particularly those exhibiting symmetry and energy functionals of limited smoothness. Its deployment in (Figueiredo et al., 2015) exemplifies its ongoing centrality within the calculus of variations and the study of nonlinear inclusions in generalized function spaces.