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Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth

Tripathi Vinayak Mani

Opuscula Mathematica

2025

Vol. 45, no. 4

523--542

In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem [formula] where Ω ⊂ RN is bounded domain with smooth boundary, [formula] M is a Kirchhoff coefficient and L denotes the mixed local and nonlocal operator. The weight function [formula] is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions.

Combined effects for a class of fractional variational inequalities

Deng Shengbing, Luo Wenshan, Ledesma César E. Torres

Opuscula Mathematica

2025

Vol. 45, no. 2

119--143

In this paper, we study the existence of a nonnegative weak solution to the following nonlocal variational inequality: [formula] for all v ∈ K, where s ∈ (0, 1) and M is a continuous steep potential well on RN. Using penalization techniques from del Pino and Felmer, as well as from Bensoussan and Lions, we establish the existence of nonnegative weak solutions. These solutions localize near the potential well Int(M−1(0)).

Small perturbations of critical nonlocal equations with variable exponents

Tao Lulu, He Rui, Liang Sihua

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20230266

In this article, we are concerned with the following critical nonlocal equation with variable exponents: [wzór], where Ω⊂RN is a bounded domain with Lipschitz boundary, N ≥ 2 , p ∈ C (Ω×Ω) is symmetric, f : C(Ω×R)→R is a continuous function, and λ is a real positive parameter. We also assume [wzór] is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations (λ small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function f does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of s = 1 and subcritical case.