Potential systems with singular $Φ$-Laplacian
Petru Jebelean
TL;DR
The paper studies solvability of a boundary value problem with a singular φ-Laplacian $- lat [ φ(u') ]' = ∇_u F(t,u)$ under multivalued boundary conditions $( φ(u')(0), -φ(u')(T)) ∈ ∂j(u(0),u(T))$. It first develops an auxiliary existence-uniqueness result for a related problem using maximal monotone operator theory, then builds a variational formulation within Szulkin’s framework for convex, lower semicontinuous perturbations to obtain minimum-energy and saddle-point solutions. A key contribution is the introduction of an eigenvalue-like constant $λ_1$ that guarantees (PS) and minimization when positive, along with universal existence results for various boundary data (including periodic forcing) and a detailed treatment of semiccoercive cases via saddle-point methods. The results extend the theory of singular φ-Laplacians to a broad class of boundary conditions and perturbations, providing robust existence results and a variational mechanism for both minimum-energy and saddle-point solutions, with concrete examples and an Appendix addressing a geometric normal-cone property.
Abstract
We are concerned with solvability of the boundary value problem $$-\left[ φ(u^{\prime}) \right] ^{\prime}=\nabla_u F(t,u), \quad \left ( φ\left( u^{\prime }\right)(0), -φ\left( u^{\prime }\right)(T)\right )\in \partial j(u(0), u(T)),$$ where $φ$ is a homeomorphism from $B_a$ -- the open ball of radius $a$ centered at $0_{\mathbb{R}^N},$ onto $\mathbb{R}^N$, satisfying $φ(0_{\mathbb{R}^N})=0_{\mathbb{R}^N}$, $φ=\nabla Φ$, with $Φ: \overline{B}_a \to (-\infty, 0]$ of class $C^1$ on $B_a$, continuous and strictly convex on $\overline{B}_a.$ The potential $F:[0,T] \times \mathbb{R}^N \to \mathbb{R}$ is of class $C^1$ with respect to the second variable and $j:\mathbb{R}^N \times \mathbb{R}^N \rightarrow (-\infty, +\infty]$ is proper, convex and lower semicontinuous. We first provide a variational formulation in the frame of critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals. Then, taking the advantage of this key step, we obtain existence of minimum energy as well as saddle-point solutions of the problem. Some concrete illustrative examples of applications are provided.