TY  - JOUR
ID  - Deng2025
LB  - Deng2025
AU  - Deng, Shengbing
AU  - Luo, Wenshan
AU  - Ledesma, César E. Torres
TI  - Combined effects for a class of fractional variational inequalities
ST  - Combined effects for a class of fractional variational inequalities
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2025
VL  - 45
IS  - 2
SP  - 119
EP  - 143
DO  - https://doi.org/10.7494/OpMath.2025.45.2.119
UR  - https://doi.org/10.7494/OpMath.2025.45.2.119
AB  - 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 \(\operatorname{Int}(M^{-1}(0))\).
KW  - fractional variational inequality
KW  - variational methods
KW  - critical nonlinearity
SN  - 1232-9274
ER  -