Maximal $$L^q$$-Regularity for Parabolic Hamilton–Jacobi Equations and Applications to Mean Field Games

In this paper we investigate maximal $$L^q$$ L q -regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided.

Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported

The aim of the present paper is to consider a fully elastic beam equation with left-end fixed and right-end simply supported, i.e., u 4 t = f t , u t , u ′ t , u ″ t , u ‴ t , t ∈ 0,1 u 0 = u ′ 0 = u 1 = u ″ 1 = 0 , where f : 0,1 × ℝ 4 ⟶ ℝ is a continuous function. By applying Leray–Schauder fixed point theorem of the completely continuous operator, the existence and uniqueness of solutions are obtained under the conditions that the nonlinear function satisfies the linear growth and superlinear growth. For the case of superlinear growth, a Nagumo-type condition is introduced to limit that f t , x 0 , x 1 , x 2 , x 3 is quadratical growth on x 3 at most.

A singular periodic Ambrosetti–Prodi problem of Rayleigh equations without coercivity conditions

In this paper, we use the Leray–Schauder degree theory to study the following singular periodic problems: [Formula: see text], [Formula: see text], where [Formula: see text] is a continuous function with [Formula: see text], function [Formula: see text] is continuous with an attractive singularity at the origin, and [Formula: see text] is a constant. We consider the case where the friction term [Formula: see text] satisfies a local superlinear growth condition but not necessarily of the Nagumo type, and function [Formula: see text] does not need to satisfy coercivity conditions. An Ambrosetti–Prodi type result is obtained.

Periodic positive solutions of superlinear delay equations via topological degree

We extend to delay equations recent results obtained by G. Feltrin and F. Zanolin for second-order ordinary equations with a superlinear term. We prove the existence of positive periodic solutions for nonlinear delay equations − u ″( t ) = a ( t ) g ( u ( t ), u ( t − τ )). We assume superlinear growth for g and sign alternance for a . The approach is topological and based on Mawhin’s coincidence degree. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Multiplicity of Solutions for Generalized Quasilinear Schrödinger Equations

In this paper, we study the following quasilinear Schrödinger equations: (P) where are given potentials, is a small parameter, g is a even function with and for all and satisfies superlinear growth at infinity. We get the existence results of multiplicity of nontrivial solutions for problem

High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equationu^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x)\bigr{)}g(u)% =0,where {\lambda,\mu>0} are parameters, {c\in\mathbb{R}}, {a(x)} is a locally integrable P-periodic sign-changing weight function, and {g\colon{[0,1]}\to\mathbb{R}} is a continuous function such that {g(0)=g(1)=0}, {g(u)>0} for all {u\in{]0,1[}}, with superlinear growth at zero. A typical example for {g(u)}, that is of interest in population genetics, is the logistic-type nonlinearity {g(u)=u^{2}(1-u)}. Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of {a(x)}. More precisely, when m is the number of intervals of positivity of {a(x)} in a P-periodicity interval, we prove the existence of {3^{m}-1} non-constant positive P-periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly non-periodic) bi-infinite sequences of three symbols.