arXiv subjects
Publications and source records attributed to Biagio Ricceri.
In this paper, on a bounded domain $Ω\subset {\bf R}^n$, we consider a nonlocal problem of the type $$\cases{-Δu=Q(u,λ)f(u) & in $Ω$ \cr & \cr u=0 & on $\partialΩ$\cr}$$ where $Q:H^1_0(Ω)\times {\bf R}\to {\bf R}$, proving, under suitable assumptions, the existence of at least three weak solutions for each $λ$ running in a suitable interval.
Let $(T,{\cal F},μ)$ be a $σ$-finite measure space, $E$ a separable real Banach space and $p\geq 1$. Given a sequence of functions $f, f_1, f_2,...$ from $T\times E$ to ${\bf R}$, under general assumptions, we prove that, for each closed hyperplane $V$ of $L^p(T,E)$, for each $u\in V$, and for each sequence $\{λ_n\}$ converging to $\int_Tf(t,u(t))dμ$, there exists a sequence $\{u_n\}$ in $V$ converging to $u$ and such that $\int_Tf_n(t,u_n(t))dμ=λ_n$ for all $n$ large enough.
Using as a main tool our recent result on the strict minimax inequality proved in [5], in this note we establish a multiplicity theorem for a problem of the type $$\cases{-K\left(\int_Ω|\nabla u(x)|^2dx\right)Δu = h(x,u) & in $Ω$\cr & \cr {{\partial u}\over {\partialν}}=0 & on $\partialΩ$.\cr}$$
In this note, we establish a multiplicity theorem for a nonlocal discrete problem of the type $$\cases{-\left(a\sum_{m=1}^{n+1}|x_m-x_{m-1}|^2+b\right)(x_{k+1}-2x_k+x_{k-1})=h_k(x_k)\hskip 10pt k=1,...,n, \cr & \cr x_0=x_{n+1}=0\cr}$$ assuming $a>0$ and (for the first time) $b<0$.
In this paper, we apply our minimax theory ([4], [5], [6]) with the one developed by A. Moameni in [2] to formalize a general scheme giving the multiplicity of critical points. Here is a sample of application of the scheme to a critical elliptic problem: Let $Ω\subset {\bf R}^n$ ($n\geq 3$) be a smooth bounded domain and let $1 0$, there exists $λ^*>0$ with the following property: for every $λ\in ]0,λ^*[$, $μ\in ]-λ^*,λ^*[$, and for every convex dense set $S\subset H^{-1}(Ω)$, there exists $\tildeφ\in S$, with $\|\tildeφ\|_{H^{-1}(Ω)}<r$, such that the problem $$\cases{-Δu=λ(|u|^{{{4}\over {n-2}}}u+ν|u|^{q-2}u+μ|u|^{p-2}u+\tildeφ) & in $Ω$\cr & \cr u=0 & on $\partialΩ$\cr}$$ has at least two solutions whose norms in $H^1_0(Ω)$ are less than or equal to $r$.
Let $H$ be a real Hilbert space and $Φ:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $Φ^{-1}(0)\neq \emptyset$ if and only if, for each $ε>0$, there exist a convex set $X\subset H$ and a convex function $ψ:X\to {\bf R}$ such that $\sup_{x\in X}(\|x\|^2+ψ(x))-\inf_{x\in X}\|x\|^2+ψ(x))<ε$ and $0\in \overline{conv}(Φ(X))$.
In this paper, given a topological space $X$, an interval $I\subseteq {\bf R}$ and five continuous functions $φ, ψ, ω:X\to {\bf R}$, $α, β:I\to {\bf R}$, we are interested in the infimum of the function $Φ:X\to ]-\infty,+\infty]$ defined by $$Φ(x)=\sup_{λ\in I}(α(λ)φ(x)+β(λ)ψ(x))+ω(x)\ .$$ Using a recent minimax theorem ([5]), we build a general scheme which provides the exact value of $\inf_XΦ$ for a large class of functions $Φ$. When additional compactness conditions are satisfied, our scheme provides also the existence of (explicitly detected) functions $γ, η:X\to {\bf R}$ such that, for some $\tilde x\in X$, one has $$γ(\tilde x)φ(\tilde x)+η(\tilde x)ψ(\tilde x)+ω(\tilde x)=\inf_{x\in X}(γ(\tilde x)φ(x)+η(\tilde x)ψ(x)+ω(x))\ .$$
In this paper, in particular, we prove the following result: Let $E$ be a reflexive real Banach space and let $C\subset E$ be a closed convex set, with non-empty interior, whose boundary is sequentially weakly closed and non-convex. Then, for every function $φ:\partial C\to {\bf R}$ and for every convex set $S\subseteq E^*$ dense in $E^*$, there exists $\tildeγ\in S$ having the following property: for every strictly convex lower semicontinuous function $J:C\to {\bf R}$, Gâteaux differentiable in $\hbox {int}(C)$, such that $J_{|\partial C}-φ$ is constant in $\partial C$ and $\lim_{\|x\|\to +\infty}{{J(x)}\over {\|x\|}}=+\infty$ if $C$ is unbounded, $\tildeγ$ is an algebraically interior point of $J'(\hbox {int}(C))$ (with respect to $E^*$).
Let $Ω\subset {\bf R}^n$ be a smooth bounded domain. In this paper, we prove a result of which the following is a by-product: Let $q\in ]0,1[$, $α\in L^{\infty}(Ω)$, with $α>0$, and $k\in {\bf N}$. Then, the problem $$\cases {-\tan\left(\int_Ω|\nabla u(x)|^2dx\right)Δu= α(x)u^q & in $Ω$\cr & \cr u>0 & in $Ω$\cr & \cr u=0 & on $\partial Ω$ \cr & \cr (k-1)π<\int_Ω|\nabla u(x)|^2dx<(k-1)π+{π\over {2}} \cr}$$ has a unique weak solution $\tilde u$ which is the unique global minimum in $H^1_0(Ω)$ of the functional $$u\to {{1}\over {2}}\tan\left (\int_Ω|\nabla\tilde u(x)|^2dx\right)\int_Ω|\nabla u(x)|^2dx-{{1}\over {q+1}}\int_Ωα(x)|u^+(x)|^{q+1}dx\ ,$$ where $u^+=\max\{0,u\}$.
Here is a sample of the results proved in this paper: Let $f:{\bf R}\to {\bf R}$ be a continuous function, let $ρ>0$ and let $ω:[0,ρ[\to [0,+\infty[$ be a continuous increasing function such that $\lim_{ξ\to ρ^-}\int_0^ξω(x)dx=+\infty$. Consider $C^0([0,1])\times C^0([0,1])$ endowed with the norm $$\|(α,β)\|=\int_0^1|α(t)|dt+\int_0^1|β(t)|dt\ .$$ Then, the following assertions are equivalent: $(a)$ the restriction of $f$ to $\left [-{{\sqrtρ}\over {2}},{{\sqrtρ}\over {2}}\right ]$ is not constant; $(b)$ for every convex set $S\subseteq C^0([0,1])\times C^0([0,1])$ dense in $C^0([0,1])\times C^0([0,1])$, there exists $(α,β)\in S$ such that the problem $$\cases{-ω\left(\int_0^1|u'(t)|^2dt\right)u"=β(t)f(u)+α(t) & in $[0,1]$\cr & \cr u(0)=u(1)=0\cr & \cr \int_0^1|u'(t)|^2dt<ρ\cr}$$ has at least two classical solutions.
This is our third paper, after [4] and [5], about a joint application of the theory developed by Brezis and Mawhin in [1] with our minimax theorems ([2], [3]) to get multiple solutions of problems of the type $$\cases{(ϕ(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\cr}$$ which are global minima of a suitable functional over a set of Lipschitzian functions. A challenging conjecture is also formulated.
In this paper, we establish an improved version of a saddle point theorem ([4]) removing a weak lower semicontinuity assumption at all. We then revisit some of the applications of that theorem in the light of such an improvement. For instance, we obtain the following very general result of local nature: Let $(H,\langle\cdot,\cdot\rangle)$ be a real Hilbert space and $Φ:B_ρ\to H$ a $C^{1,1}$ function, with $Φ(0)\neq 0$. Then, for each $r>0$ small enough, there exist only two points points $x^*, u^*\in S_r$, such that $$\max\{\langle Φ(x^*),x^*-x\rangle, \langle Φ(x),x^*-x\rangle\}< 0\ ,$$ for all $x\in B_r\setminus \{x^*\}$, $$\|Φ(u^*)-u^*\|=dist(Φ(u^*),B_r)$$ and $$\|Φ(x)-u^*\|<\|Φ(x)-x\|$$ for all $x\in B_r\setminus \{u^*\}$, where $$B_r=\{x\in H : \|x\|\leq r\}$$ and $$S_r=\{x\in H : \|x\|=r\}\ .$$
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 normed space $(E,\|\cdot\|)$ and let $Y$ be a convex subset of $E$ such that $X\subseteq \overline {Y}$. Then, for every convex set $S\subseteq Y$ dense in $Y$, for every upper semicontinuous bounded function $γ:X\to {\bf R}$ and for every $λ>{{4\sup_X|γ|}\over {diam(X)}}$, there exists $y^*\in S$ such that the function $x\to γ(x)+λ\|x-y^*\|$ has at least two global maxima in $X$.
In this paper, given two Banach spaces $X, Y$ and a $C^1$ functional $Φ:X\times Y\to {\bf R}$, under general assumptions, we show that either $Φ$ has a saddle-point in $X\times Y$ or, for each convex and dense set $S\subseteq Y$, there is some $\tilde y\in S$ such that $Φ(\cdot,\tilde y)$ has at least three critical points in $X$, two of which are global minima. Also, an application to non-cooperative elliptic systems is presented.
We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $Ω\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $Φ:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $Φ(0,0)=0$, such that $$\sup_{(u,v)\in {\bf R}^2}{{|Φ_u(u,v)|+|Φ_v(u,v)|}\over {1+|u|^p+|v|^p}}<+\infty$$ where $p>0$, with $p={{2}\over {n-2}}$ when $n>2$. Then, for every convex set $S\subseteq L^{\infty}(Ω)\times L^{\infty}(Ω)$ dense in $L^2(Ω)\times L^2(Ω)$, there exists $(α,β)\in S$ such that the problem $$\cases {-Δu=(α(x)\cos(Φ(u,v))-β(x)\sin(Φ(u,v)))Φ_u(u,v) & in $Ω$ \cr & \cr -Δv= (α(x)\cos(Φ(u,v))-β(x)\sin(Φ(u,v)))Φ_v(u,v) & in $Ω$ \cr & \cr u=v=0 & on $\partialΩ$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Ω)\times H^1_0(Ω)$ of the functional $$(u,v)\to {{1}\over {2}}\left ( \int_Ω|\nabla u(x)|^2dx+\int_Ω|\nabla v(x)|^2dx\right )$$ $$-\int_Ω(α(x)\sin(Φ(u(x),v(x)))+β(x)\cos(Φ(u(x),v(x))))dx\ .$$
Here is one of the results obtained in this paper: Let $Ω\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $λ_1$ be the first eigenvalue of the problem $$\cases{-Δu=λu & in $Ω$ \cr & \cr u=0 & on $\partialΩ$\ .\cr}$$ Then, for every $λ>λ_1$ and for every convex set $S\subseteq L^{\infty}(Ω)$ dense in $L^2(Ω)$, there exists $α\in S$ such that the problem $$\cases{-Δu=λ(u^+-(u^+)^q)+α(x) & in $Ω$ \cr & \cr u=0 & on $\partialΩ$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Ω)$ of the functional $$u\to {{1}\over {2}}\int_Ω|\nabla u(x)|^2dx-λ\int_Ω\left ({{1}\over {2}}|u^+(x)|^2-{{1}\over {q+1}}|u^+(x)|^{q+1}\right )dx-\int_Ωα(x)u(x)dx\ $$ where $u^+=\max\{u,0\}$.
In this very short paper, we provide a strong motivation for the study of the following problem: given a real normed space $E$, a closed, convex, unbounded set $X\subseteq E$ and a function $f:X\to X$, find suitable conditions under which, for each $y\in X$, the function $$x\to \|x-f(x)\|-\|y-f(x)\|$$ has at most one global minimum in $X$.
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 $Φ:B\to H$ a $C^{1,1}$ function, with $Φ(0)\neq 0$, such that the function $x\to \langle Φ(x),x-y\rangle$ is weakly lower semicontinuous in $B$ for all $y\in B$. Then, for each $r>0$ small enough, there exists a unique point $x^*\in H$, with $\|x^*\|=r$, such that $$\max\{\langle Φ(x^*),x^*-y\rangle, \langle Φ(y),x^*-y\rangle\}< 0$$ for all $y\in H\setminus \{x^*\}$, with $\|y\|\leq r$.