Paths to uniqueness of critical points and applications to partial differential equations

Abstract

We prove a unified and general criterion for the uniqueness of critical points of a functional in the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and significantly generalizes well-known uniqueness theorems. Due to the flexibility in the construction of the paths, our approach does not depend on the convexity of the domain and can be used to prove uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, thus they provide uniqueness of solutions to the corresponding Euler-Lagrange equations. For functionals emerging from elliptic problems, the assumptions of our abstract theorems follow from maximum principles, decay properties, and novel general inequalities. To illustrate our method we present a unified proof of known results, as well as new theorems for mean-curvature type operators, fractional Laplacians, Hamiltonian systems, Schrödinger equations, and Gross-Pitaevski systems.

On the rate of change of the sharp constant in the Sobolev-Poincaré inequality

“…1 which implies(1.8). Moreover, equality in (6.1) forces equality in (4.2), which in turn forces Ω to be a round ball.…”

“…wherẽis an extremal function for  , in the case of the unit ball 1 . Equality can only occur if Ω is a round ball.…”

“…For , there is a unique positive minimizer ϕ normalized by (see Franzina and Lamberti [, Theorem 3.1] or the work of Bonheure et al. [, Examples 4.4 and 4.6] for the case while uniqueness follows from the maximum principle in the case ). We therefore restrict ourselves to p in the range .…”

Some uniqueness results in quasilinear subhomogeneous problems

“…A rather general criterion has been recently formulated in [6] to prove uniqueness results for positive critical points of a given functional. Roughly speaking, the authors take advantage of a basic convexity principle, namely, the fact that if f : [0, 1] → R is differentiable and strictly convex then it has at most one critical point.…”

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We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in [6]. We apply it to generalized p-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions.

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