Special Session 33: Variational, Topological and Set-Valued Methods for Nonlinear Differential Problems

The Prescribed Gauss Curvature Problem and its corresponding Flows

Franziska Borer University of Technology Berlin Germany Co-Author(s):    Esther Cabezas--Rivas, Peter Elbau, Tobias Weth, Patrick Winkert

Abstract: In mathematics, there has long been an intrinsic motivation to formulate static problems in a flow context. Typically, elliptic theories are regarded as the static case of the corresponding parabolic problem; in that sense, many times the better-understood elliptic theory has been a source of intuition to generalise the corresponding results in the parabolic case. Examples for this duality are minimal surfaces and the mean curvature flow, harmonic maps and the solutions of the heat equation, as well as the Uniformisation Theorem and the two-dimensional normalised Ricci flow. In this talk, we give a glimpse into several results regarding the prescribed Gauss curvature problem---a problem raised by Kazdan and Warner dealing with the question which smooth functions $f:M\to\R$ arise as the Gauss curvature $K_g$ of a conformal metric $g(x)=\e^{2u(x)}\bar g(x)$ on a closed Riemannian manifold $(M,\bar g)$---and its corresponding flows.

Qualitative results for elliptic systems with variable exponents

Maria-Magdalena Boureanu University of Craiova Romania Co-Author(s):    Alejandro Velez-Santiago

Abstract: We investigate anisotropic elliptic systems with variable exponents governed by Leray-Lions type operators acting both in the domain and on its boundary. Using variational methods and various tools from the variable exponent analysis, we address the weak solvability of such systems. We prove existence, uniqueness, and global regularity of weak solutions, and illustrate our framework through relevant examples. This is a joint work with Alejandro Velez-Santiago.

Symmetry breaking for elliptic equations with exponential nonlinearities

Francesca Colasuonno Universita degli Studi di Torino Italy Co-Author(s):

Abstract: Radially symmetric semilinear elliptic Dirichlet problems with exponential nonlinearities are considered in possibly unbounded domains of $\mathbb R^N$. For this class of problems, we prove the existence of a positive symmetry-breaking solution by using techniques in the spirit of Szulkin`s nonsmooth critical point theory, applied within invariant convex cones.

Anisotropic Singular Logistic Equations

Leszek Gasinski University of the National Education Commission Poland Co-Author(s):

Abstract: We consider a parametric Dirichlet problem driven by the anisotropic (p,q)-Laplacian and a reaction with a singular term and a superdiffusive logistic perturbation. We prove an existence and nonexistence theorem which is global with respect to the parameter.

The p-Laplacian obstacle problem with singular and discontinuous reaction term

Umberto Guarnotta University of Catania Italy Co-Author(s):    Annamaria Barbagallo

Abstract: The talk is devoted to discuss existence and regularity of solutions to a p-Laplacian obstacle problem whose reaction is singular (i.e., it blows up when the solution approaches zero) and possesses a null-measure set of discontinuity points. The techniques presented are based on non-smooth calculus, convex analysis, monotonicity techniques, regularity theory, and locality properties.

Concentration Phenomena and Multi-bump Structures in Logarithmic $p$-Laplacian Equations

Lin Li Chongqing Technology and Business University Peoples Rep of China Co-Author(s):    Huo Tao, Patrick Winkert

Abstract: This talk investigates a class of quasilinear Schr\{o}dinger equations involving the $p$-Laplacian operator and logarithmic nonlinearities. The presence of the singular term $u \log u^p$ renders the energy functional nonsmooth on standard Sobolev spaces, requiring the use of Szulkin's critical point theory and penalization methods.

Existence and localization results for elliptic problems with intrinsic operators

Angela Sciammetta University of Palermo Italy Co-Author(s):

Abstract: The aim of this talk is to study a class of elliptic problems and systems characterized by the presence of intrinsic operators in the reaction terms. The main goal is to establish results on the existence and localization of solutions in frameworks where the nonlinearities depend on the function, its gradient, and an intrinsic operator. The approach is based on a suitable sub- and supersolution method.

A Leray-Lions approach to logarithmic double phase problems with natural growth terms

Matheus Stapenhorst Universidade Estadual Paulista Julio de Mesquita Filho Brazil Co-Author(s):    Marcos Tadeu de Oliveira Pimenta; Patrick Winkert

Abstract: In this talk we will study problems involving the logarithmic double phase operator and a nonlinear term with natural growth. A solution to our problem is obtained as the limit of solutions to approximated problems, whose solutions exist by the Leray-Lions Theorem. The full result is obtained by carefully estimating certain terms associated to the pseudomonotonicity of the operator. This talk is based on a joint work with Marcos Tadeu de Oliveira Pimenta (Presidente Prudente, Brazil) and Patrick Winkert (Berlin, Germany).

Existence and concentration of ground state solutions for a Kirchhoff type problem involving the 1-Laplacian operator

Jijiang Sun Nanchang University Peoples Rep of China Co-Author(s):    Jiaqian Zhang and Jianjun Zhang

Abstract: In this talk, we are concerned with the existence and concentration behavior of ground state solutions for the following Kirchhoff type equations involving the 1-Laplacian operator $$\left(a+b\left(\displaystyle\int_{\mathbb R^N}\epsilon|Du|+\int_{\mathbb R^N}V(x)|u|\right)^{\alpha-1}\right)\left(-\epsilon\Delta_{1}u+V(x)\displaystyle\frac{u}{|u|}\right) = f(u)$$ in $\mathbb R^N$, where $a, b>0$, $N\geq2$, $\epsilon>0$ is a small parameter, $\alpha\in(1,\frac{N}{N-1})$, and the operator $\Delta_1$ is the well known 1-Laplacian operator. Under suitable conditions on $V$ and $f$, using non-smooth critical point theory, Lions' Concentration-Compactness Principle and some ingenious analyses, we first prove the existence of ground state solutions $u_\epsilon$ for a small parameter $\epsilon > 0$ in $BV(\mathbb R^N)$, the space of functions of bounded variation, which is the natural functional setting for the 1-Laplacian. Subsequently, we demonstrate that as $\epsilon \to 0$, this family of solutions concentrates around a global minimum of the potential $V$. This work is jointly with Jiaqian Zhang and Jianjun Zhang.

Remarks on positive solutions to a $p$-Laplacian problem with a possibly singular nonlinearity

Bruno Vassallo University of Messina Italy Co-Author(s):    Pasquale Candito and Giuseppe Failla

Abstract: In this paper, we combine variational methods and truncation techniques to study the existence of a positive weak solution for a quasilinear elliptic problem driven by the $p$-Laplacian operator involving a reaction term which might or might not have a singularity at $0$. Furthermore, provided that solutions belong to $C^1(\overline{\Omega})$, uniqueness is achieved using a D\`{i}az-Sa\`{a} type argument, which relies on a Br\`{e}zis-Oswald assumption on the nonlinearity. Additionally, in the superlinear case, we give a multiplicity result that applies when an Ambrosetti-Rabinowitz type condition is fulfilled, e.g. in the concave-convex context.

Global regularity for Robin-type double phase boundary value problems

Alejandro Velez-Santiago University of Puerto Rico - Rio Piedras Campus USA Co-Author(s):    Angel Crespo-Blanco, Patrick Winkert

Abstract: We are concerned with a inhomogeneous elliptic problem involving the double phase Laplace-type operator and Robin boundary conditions. Under minimal conditions on the coefficients and inhomogeneous data, we establish the first known result concerning the full global H\older continuity of weak solutions.