报告题目1：A quasilinear modifiled Schrodinger equation: from bound domains to the Euclidean space

报告人：Anna Maria Candela教授，Universita degli Studi di Bari Aldo Moro

报告时间：2026年10月20日（周二）8:30-12:30

报告地点：20-404

报告摘要：In the last years we have investigated the existence of solutionsof the quasilinear elliptic problem\[(P)\qquad \left\{\begin{array}{ll}- {\rm div} (A(x,u) |\nabla u|^{p-2}\nabla u) + \frac1p A_t(x,u) |\nabla u|^p\ =\ g(x,u)&\hbox{in $\Omega$,}\\u = 0 &\hbox{on $\partial\Omega$,}\end{array}\right.\] with $p > 1$, $\Omega$ open bounded domain in $\mathbb{R}^N$ ($N\ge 2$), where $A(x,t)$, $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and$g(x,t)$ are Carath\'eodory functions on $\Omega \times \mathbb{R}$.

Taking $G(x,t) = \int_0^t g(x,s) ds$, suitable assumptions on $A(x,t)$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional is\[{\cal J}(u)\ =\  \frac1p\ \int_{\Omega} A(x,u)|\nabla u|^p dx - \int_{\Omega} G(x,u) dx,\]which is $C^1$ but not verifies the classical Palais--Smale conditionon the Banach space $X = W^{1,p}_0(\Omega) \cap L^\infty(\Omega)$equipped with the intersection norm $\|\cdot\|_X$.Anyway, following an approach which exploits the interaction between$\|\cdot\|_X$ and the standard norm on $W^{1,p}_0(\Omega)$,we apply suitable generalizations of classical variational theoremsto ${\cal J}$ in $X$ so to prove the existence of weak solutionsof $(P)$ by comparing the growth of $A(x,t)|\xi|^p$ with that one of $G(x,t)$.Recently, such results have allowed us to introduce an approximating argument for thequasilinear modified Schr枚dinger equation\[-{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p+ V(x)|u|^{p-2} u = f(x,u)\quad\mbox{ in } \mathbb{R}^N,\]which admits at least one nontrivial weak bounded solutionif the potential $V(x)$satisfies ``good'' hypotheses.On the contrary, we are able to state only a dicothomy result if $V(x) \equiv 1$.\bigskip\noindentJoint works withGiuliana Palmieri, Addolorata Salvatore and Caterina Sportelli.

报告题目2：Existence results for a borderline case of a class of *p*-Laplacian problems

报告人：Anna Maria Candela教授，Universita degli Studi di Bari Aldo Moro

报告时间：2026年10月21日（周三）14:00-18:00

报告地点：20-308

报告摘要：The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically `linear' problem \[\begin{cases}-\operatorname{div}\left[\left(A_0(x)+A(x)(|u|)^{ps}\right)(|\nabla u|)^{p-2}\nabla u\right] + sA(x)(|u|)^{ps-2}u(|\nabla u|)^p \\  \quad = \mu (|u|)^{p(s+1)-2}u + g(x,u) & \text{in } \Omega, \\[0.5em]u = 0 & \text{on } \partial\Omega, \end{cases}\] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N \geq 2\), \(1 < p < N\), \(s > 1/p\), both the coefficients \(A_0(x)\) and \(A(x)\) are in \(L^\infty(\Omega)\) and far away from \(0\), \(\mu \in \mathbb{R}\), and the `perturbation' term \(g(x,t)\) is a Carath\'eodory function on \(\Omega \times \mathbb{R}\) which grows as \((|t|)^{r-1}\) with \(1 \leq r < p(s+1)\) and is such that \[g(x,t) \approx \nu(|t|)^{p-2}t \quad \text{as } t \to 0.\] By introducing suitable thresholds for the parameters \(\nu\) and \(\mu\), which are related to the coefficients \(A_0(x)\), respectively \(A(x)\), under suitable hypotheses on \(g(x,t)\), the existence of a nontrivial weak solution is proved if either \(\nu\) is large enough with \(\mu\) small enough or \(\nu\) is small enough with \(\mu\) large enough. Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used.

报告人简介：安娜·玛丽亚·坎德拉（Anna Maria Candela），意大利巴里阿尔多·莫罗大学数学系数学分析专业正教授，现任数学系主任、校学术委员会委员，担任国际期刊《Mediterranean Journal of Mathematics》编委。主要从事变分方法、拓扑方法及其在非线性微分方程中的应用研究，重点关注p-拉普拉斯型拟线性椭圆方程、拟线性修正薛定谔方程等问题解的存在性与多重性，研究还涉及半黎曼流形上的测地线与轨道问题。在Arch. Ration. Mech. Anal.、Calc. Var. Partial Differential Equations、J. Differential Equations、Commun. Contemp. Math.、Nonlinear Anal.、Proc. Roy. Soc. Edinburgh Sect. A等国际期刊发表学术论文60余篇。曾赴北京大学、南开大学陈省身数学研究所、德国柏林洪堡大学、西班牙格拉纳达大学等高校和研究机构开展学术访问，并多次应邀在国际学术会议及高校作学术报告。

邀请人：非线性分析与PDE团队