187,700 research outputs found
Optimal regularity results for the one-dimensional prescribed curvature equation via the strong maximum principle
No full textA refined version of the strong maximum principle is proven for a class
of second order ordinary differential equations with possibly discontinuous
non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity
results, recently established by L\'opez-G\'omez and Omari for the bounded variation solutions of non-autonomous quasilinear equations driven by the one-dimensional curvature operator, are substantially improved by admitting general prescribed curvatures and by incorporating general boundary conditions. The novel approach developed here yields a new, deeper, interpretation of the assumptions introduced in our previous papers, simultaneously clarifying their meaning and making fully transparent their connection with the strong maximum principle
Period two implies any period for a class of differential inclusions
No full textWe produce a detailed proof of a result stated in "F. Obersnel and P. Omari, Period two implies chaos for a class of ODEs, Proc. Amer. Math. Soc, 135 (2007), 2055-2058" concerning scalar time-periodic first order differential inclusions. Such a result shows that the existence of just one subharmonic implies the
existence of large sets of subharmonics of all given orders
Regular versus singular solutions in quasilinear indefinite problems with sublinear potentials
No full textMinisterio de Ciencia, Innovación y Universidades (España)Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasInstituto de Matemática Interdisciplinar (IMI)TRUEpu
Pairs of positive solutions of a quasilinear elliptic Neumann problem driven by the mean curvature operator
No full textWe establish the existence of multiple positive weak solutions of the quasilinear elliptic Neumann problem driven by the mean curvature operator (formula Presented) Ω. Here, Ω is a bounded regular domain in RN, with N ≥ 2, p ∈ (1, 1∗), w is a sign-changing weight function, and λ > 0 is a parameter. Our findings provide the existence, for sufficiently small λ, of two positive solutions, the smaller in C1(Ω), the larger in BV (Ω), which respectively bifurcate from (λ, u) = (0, 0) and from (λ, u) = (0, +∞). This way we extend to a genuine PDE setting some results obtained in [22, 23] for the corresponding one-dimensional problem
Infinitely Many Regular Weak Solutions for Odd Symmetric Prescribed Mean Curvature Problems
No full textWe establish in this paper the existence of infinitely many regular weak solutions of the prescribed mean curvature problem
\begin{equation*}
-{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big)
= f(x,u)
\mbox{\, in },
\qquad
\mathcal{B} u=0 \mbox{\, on }.
\end{equation*}
where is a bounded domain in \RR^N with a boundary , \BB is either the Dirichlet, or the Neumann, or the mixed boundary operator, the function is odd with respect to s\in \RR and has a potential which is desultorily subquadratic at , locally with respect to . Our findings improve and extend in various directions previous results established in the literature
PERIODIC BVPS AT RESONANCE: A GLANCE AT SOME OF FABIO ZANOLIN'S MANIFOLD ACHIEVEMENTS
No full textDuring his career Fabio Zanolin has obtained many relevant results about the existence, the nonexistence, and the multiplicity of periodic solutions for ordinary differential equations at resonance. A small part of them is here surveyed, weaving the presentation with a few personal memories
Positive solutions of superlinear indefinite prescribed mean curvature problems
No full textThis paper analyzes the superlinear indefinite prescribed mean curvature problem [ -mathrm{div}left({ abla u}/{sqrt{1+| abla u|^2}} ight)=lambda a(x)h(u) quad ext{in }Omega,qquad u=0 quad ext{on } partialOmega, ] where is a bounded domain in with a regular boundary , satisfies , as , being an exponent with represents a parameter, and is a sign-changing function. The main result establishes the existence of positive regular solutions when is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for small is further discussed assuming that satisfies as , being such that if ; thus, in dimension , the function is not superlinear at , although its potential is. Imposing such different degrees of homogeneity of at and at is dictated by the specific features of the mean curvature operator
Nonresonance conditions on the potential for e second-order periodic boundary value problem
No full textCharacterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operator
No full textThe aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem egin{equation*} label{P} -(u'/sqrt{1+(u')^2})' = lambda a(x) f(u) ; ; ext{in } (0,1), quad u'(0)=0,;u'(1)=0, end{equation*} where is a parameter, changes sign once in at the point , and is positive and increasing in with a potential, , superlinear at . oindent In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as , we can characterize the existence of singular bounded variation solutions solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition medskip egin{center} quad and quad end{center} medskip No previous result of this nature is known in the context of the theory of superlinear indefinite problems
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