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- ItemSome useful tools in the study of nonlinear elliptic problems(Elsevier, 2024-12-05) Papageorgiou, Nikolaos S.; Radulescu, VicentiuThis paper gives an overview of some basic aspects concerning the qualitative analysis of nonlinear, nonhomogeneous elliptic problems. We are concerned with two classes of elliptic equations with Dirichlet boundary condition. The first problem is driven by a general nonhomogeneous differential operator, which includes several usual operators (such as the (p,q)-Laplace operator introduced by P. Marcellini). Next, we focus on differential operators with unbalanced growth in the nonautonomous case. Our analysis will point out some relevant differences between balanced and unbalanced growth problems. The presentation is done in the context of Dirichlet problems but a similar analysis can be developed for other boundary conditions, such as Neumann or Robin.
- ItemUniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential(The French Academy of sciences, 2025-06-02) Tianxiang, Gou; Radulescu, VicentiuIn this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential: (MATHEMATICAL FOMULA PRESENTED) where (MATHEMATICAL EQUATION PRESENTED) is the lowest eigenvalue of the operator ()s + |x|2. This solves an open question raised in [15] concerning the uniqueness of solutions to the equation.
- ItemGlobal Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions(Springer Nature, 2025-03-26) Zhang, Jian; Radulescu, Vicentiu; Yang, Minbo; Zhou, JiazhengIn this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy-Littlewood-Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for J(mu)(u0) < 0 and decay rate of H-0(1) and L-2-norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.
- ItemPlanar Choquard equations with critical exponential reaction and Neumann boundary condition(Wiley, 2024-10-26) Rawat, Sushmita; Radulescu, Vicentiu; Sreenadh, KonijetiWe study the existence of positive weak solutions for the following problem: (Formula presented.) where (Formula presented.) is a bounded domain in (Formula presented.) with smooth boundary, (Formula presented.) is a bounded measurable function on (Formula presented.), (Formula presented.) is nonnegative real number, (Formula presented.) is the unit outer normal to (Formula presented.), (Formula presented.), and (Formula presented.). The functions (Formula presented.) and (Formula presented.) have critical exponential growth, while (Formula presented.) and (Formula presented.) are their primitives. The proofs combine the constrained minimization method with energy methods and topological tools.
- ItemOn solvability of a two-dimensional symmetric nonlinear system of difference equations(Springer, 2024-08-27) Stevič, Stevo; Iričanin, Bratislav; Kosmala, Witold; Šmarda, ZdeněkWe show that the system of difference equations (Formula presented.) where $kN, lN_0, l