Opuscula Math. 45, no. 2 (2025), 119-143
https://doi.org/10.7494/OpMath.2025.45.2.119

 
Opuscula Mathematica

Combined effects for a class of fractional variational inequalities

Shengbing Deng
Wenshan Luo
César E. Torres Ledesma

Abstract. In this paper, we study the existence of a nonnegative weak solution to the following nonlocal variational inequality: \[\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} u (-\Delta)^{{\frac{s}{2}}}(v-u)dx+\int_{\mathbb{R}^N}(1+\lambda M(x))u(v-u)dx \geq \int_{\mathbb{R}^N}f(u)(v-u)dx, \] for all \(v \in\mathbb{K}\), where \(s\in (0,1)\) and \(M\) is a continuous steep potential well on \(\mathbb{R}^N\). 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 \(\operatorname{Int}(M^{-1}(0))\).

Keywords: fractional variational inequality, variational methods, critical nonlinearity.

Mathematics Subject Classification: 35A15, 35J86, 49J40.

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  • César E. Torres Ledesma (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-6889-4982
  • FCA Research Group, Departamento de Matemáticas, Instituto de Investigación en Matemáticas, Faculta de Ciencias Físicas y Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Trujillo, Perú
  • Communicated by Vicenţiu D. Rădulescu.
  • Received: 2024-09-23.
  • Revised: 2025-01-21.
  • Accepted: 2025-01-22.
  • Published online: 2025-03-10.
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Cite this article as:
Shengbing Deng, Wenshan Luo, César E. Torres Ledesma, Combined effects for a class of fractional variational inequalities, Opuscula Math. 45, no. 2 (2025), 119-143, https://doi.org/10.7494/OpMath.2025.45.2.119

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