• Corpus ID: 211020583

# An alternative theorem for gradient systems.

@article{Ricceri2020AnAT,
title={An alternative theorem for gradient systems.},
author={Biagio Ricceri},
journal={arXiv: Analysis of PDEs},
year={2020}
}
• B. Ricceri
• Published 4 February 2020
• Mathematics, Physics
• arXiv: Analysis of PDEs
Here is one of the result obtained in this paper: Let $\Omega\subset {\bf R}^2$ be a smooth bounded domain and let $F, G : {\bf R}\to {\bf R}$ be two $C^1$ functions satisfying the following conditions: $(i)$ for some $p>0$, one has $$\limsup_{|\xi|\to +\infty}{{|F'(\xi)|+|G'(\xi)|}\over {|\xi|^p}}<+\infty\ ;$$ $(ii)$ $F$ is non-negative, non-decreasing, $\lim_{\xi\to +\infty}{{F(\xi)}\over {\xi^2}}=0$, $\lim_{\xi\to 0^+}{{F(\xi)}\over {\xi^2}}=+\infty$ and the function $\xi\to {{F'(\xi)}\over… 3 Citations A class of functionals possessing multiple global minima We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let$\Omega\subset {\bf R}^n$($n\geq 2$) be a smooth bounded domain and let$\Phi:{\bf R}^2\to {\bf R}$be A more complete version of a minimax theorem In this paper, we present a more complete version of the minimax theorem established in [7]. As a consequence, we get, for instance, the following result: Let X be a compact, not singleton subset of A Class of Equations with Three Solutions Here is one of the results obtained in this paper: Let$\Omega\subset {\bf R}^n$be a smooth bounded domain, let$q>1$, with$q \lambda_1$and for every convex set$S\subseteq L^{\infty}(\Omega)$## References SHOWING 1-10 OF 29 REFERENCES On the Infimum of Certain Functionals Here is a particular case of our main result: Let X be a real Banach space, $$\varphi: X \rightarrow \mathbf{R}$$ a nonzero continuous linear functional and ψ: X → R a nonconstant Lipschitzian Applying twice a minimax theorem Here is one of the results obtained in this paper: Let$X, Y$be two convex sets each in a real vector space, let$J:X\times Y\to {\bf R}$be convex and without global minima in$X$and concave in A remark on variational inequalities in small balls In this paper, we prove the following result: Let$(H,\langle\cdot,\cdot\rangle)$be a real Hilbert space,$B$a ball in$H$centered at$0$and$\Phi:B\to H$a$C^{1,1}$function, with$\Phi(0)\neq
Energy functionals of Kirchhoff-type problems having multiple global minima
In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let $\Omega\subset {\bf R}^n$ be a smooth
A minimax theorem in infinite-dimensional topological vector spaces
In this paper, we obtain a minimax theorem by means of which, in turn, we prove the following result: Let $E$ be an infinite-dimensional reflexive real Banach space, $T:E\to E$ a non-zero compact
Well-posedness of constrained minimization problems via saddle-points
• B. Ricceri
• Mathematics, Computer Science
J. Glob. Optim.
• 2008
A very general well-posedness result is proved for a class of constrained minimization problems of which the following is a particular case: Let X be a Hausdorff topological space and J, Phi: X be two non-constant functions such that J has sequentially compact sub-level sets and admits a unique global minimum in X.
Strongly indefinite functionals and multiple solutions of elliptic systems
• Mathematics
• 2003
We study existence and multiplicity of solutions of the elliptic system \( \left\{{\begin{array}{*{20}l} {- \Updelta u = H_{u} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,} \hfill \\ {- \Updelta v =
A strict minimax inequality criterion and some of its consequences
In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is
Sublevel sets and global minima of coercive functionals and local minima of their perturbations
The aim of the present paper is essentially to prove that if $\Phi$ and $\Psi$ are two sequentially weakly lower semicontinuous functionals on a reflexive real Banach space and if $\Psi$ is also
Another multiplicity result for the periodic solutions of certain systems
In this paper, we deal with a problem of the type $$\cases{(\phi(u'))'=\nabla_xF(t,u) & in [0,T]\cr & \cr u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\ ,\cr}$$ where, in particular, $\phi$ is a