Weak solutions for the fractional Kirchhoff-type problem via Young measures
Main Article Content
Abstract
The aim of this paper is to investigate the existence of weak solutions to the following Kirchhoff-type problem:
$$\begin{cases}
M\left([u]_{s p}^p\right) (-\Delta)_p^s (u)=f(x,u) \quad &\text { in } \Omega,\\ u=0 \quad &\text { in } \mathbb{R}^n \backslash \Omega,
\end{cases}$$
where $\Omega\subset\mathbb{R}^n$, $0<s<1<p<\infty$, $[u]_{s p}$ is Gagliardo semi-norm, $M$ is a continuous function with value in $\mathbb{R}^+$, $f$ a given function and $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator.
Under appropriate assumptions on the main functions, we
obtain the existence results by applying the Galerkin method combined with the theory of Young measures.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
R. A. Adams and J. J. Fournier, Sobolev spaces, Elsevier, 2003.
C. O. Alves, F. J. S. A. Corrˆea and G. M. Figueiredo, On a class of nonlocal elliptic problems
with critical growth, Differ. Equ. Appl., 2(2010), 409-417.
L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter
functionals, Manuscripta Mathematica, 134(2011), 377-403.
G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear
Analysis: Theory, Methods and Applications, 73(2010), 1952-1965.
G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems,
Archive for rational mechanics and analysis, 196(2010), 489-516.
E. Azroul, A. Benkirane and M, srati, Three solutions for a Kirchhoff-type problem involving
nonlocal fractional p-Laplacian. In Advances in Operator Theory, 4(2019), 821–835.
E. Azroul and F. Balaadich, Existence of solutions for a class of Kirchhoff-type equation via
Young measures, Numerical Functional Analysis and Optimization, 42(2021), 460-473.
E. Azroul F. Balaadich, On strongly quasilinear elliptic systems with weak monotonicity,
Journal of Applied Analysis, 27(2021), 153-162.
E. Azroul and F. Balaadich, A weak solution to quasilinear elliptic problems with perturbed
gradient, Rendiconti del Circolo Matematico di Palermo Series 2, 70(2021), 151-166
F. Balaadich and E. Azroul Quasilinear elliptic systems in perturbed form, International
Journal of Nonlinear Analysis and Applications, 10(2019), 255-266.
F. Balaadich and E. Azroul, Elliptic Systems of p-Laplacian Type, Tamkang Journal of
Mathematics, 53(2022), 11-21.
F. Balaadich and E. Azroul, Existence results for fractional p-Laplacian systems via young
measures, Mathematical Modelling and Analysis, 27(2022), 232-241.
B. Barrios, E. Colorado, A. De Pablo and U. S´anchez, On some critical problems for the
fractional Laplacian operator, Journal of Differential Equations, 252(2012), 6133-6162.
G. M. Bisci, Fractional equations with bounded primitive, Applied Mathematics Letters,
(2014), 53-58.
G. M. Biscia and D. Repovˇs, Higher nonlocal problems with bounded potential, Journal of
Mathematical Analysis and Applications, 420(2014), 167-176.
L. A. Caffarelli, J. M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Communications
on Pure and Applied Mathematic, 63(2010), 1111-1144.
X. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional
Laplacian with general nonlinearity, Nonlinearity, 26(2013), 479.
W. Chen and S. Deng, Existence of solution for a Kirchhoff type problem involving the fractional
p-Laplace operator, Electronic Journal of Qualitative Theory of Differential Equations,
(2015), 1-8.
K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical
Analysis and Applications, 265(2002), 229-248.
G. Dolzmann, N. Hungerb¨uhler and S. M¨uller, Non-linear elliptic systems with measurevalued
right hand side, Math. Z., 226(1997), 545–574.
G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical
growth via truncation argument, Journal of Mathematical Analysis and Applications,
(2013), 706-713.
A. Fiscella and R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces,
Ann. Acad. Sci. Fenn. Math., 40(2015), 235-253.
A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator,
Nonlinear Analysis: Theory, Methods and Applications, 94(2014), 156-170.
N. Hungerb¨uhler, A refinement of Ball’s theorem on Young measures, New York J. Math.,
(1997), 53.
A. Iannizzotto and M. Squassina, 12-Laplacian problems with exponential nonlinearity, Journal
of Mathematical Analysis and Applications, 414(2014), 372-385.
F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems
via critical point theory, Computers Mathematics with Applications, 62(2011), 1181-1199.
G. Kirchhoff, Mechanik, Leipzig: Teubner, 1883.
J.L. Lions, Quelques m´ethodes de r´esolution de probl`emes aux limites non lin´eaires, ´Etudes
math´ematiques, 1969.
Y. Liu, Z. Liu, C. F. Wen and J. C. Yao, Existence of solutions for non-coercive variationalhemivariational
inequalities involving the nonlocal fractional p-Laplacian, Optimization,
(2022), 485-503.
X. Mingqi, G. M. Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary
Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity, 29(2016), 357.
K. Ho, K. Perera, I. Sim and M. Squassina, A note on fractional p-Laplacian problems with
singular weights, Journal of Fixed Point Theory and Applications, 19(2017), 157-173.
H. Qiu and M. Xiang, Existence of solutions for fractional p-Laplacian problems via Leray-
Schauder’s nonlinear alternative, Boundary Value Problems, 2016(2016), 1-8.
R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, Journal
of Mathematical Analysis and Applications, 389(2012), 887-898.
I. Talibi, B. El Boukari and J El Ghordaf, Existence of weak solutions to the fractional
p-Laplacian problems of Kirchhoff-type via topological degree, Boletim da Sociedade
Paranaense de Matematica, 43(2025), 1-10.
I. Talibi, F. Balaadich, B. El Boukari, and J. El Ghordaf, On the fractional p-Laplacian
problem via Young measures, Khayyam Journal of Mathematics, 10(2024) (2), 337-348.
I. Talibi, B. El Boukari, J. El Ghordaf, A. Taqbibt and F. Balaadich, Kirchhoff-type problem
involving the fractional p-Laplacian via Young measures, Filomat, 38 (2024), 8211-8223.
I. Talibi, F. Balaadich, B. El Boukari, and J. El Ghordaf, On weak solutions to parabolic
problem involving the fractional p-Laplacian via Young measures, In Annales Mathematicae
Silesianae, (2024).
K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic
operators, Nonlinear Analysis: Real World and Applications, 14(2013), 867-874.
M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving
the non-local fractional p-Laplacian, Journal of Mathematical Analysis and Applications,
(2015), 1021-1041.
M. Xiang, B. Zhang and V. D. R˘adulescu, Multiplicity of solutions for a class of quasilinear
Kirchhoff system involving the fractional p-Laplacian, Nonlinearity, 29(2016), 3186.
J. Zhang, D. Yang and Y. Wu, Existence results for a Kirchhoff-type equation involving
fractional p(x)-Laplacian, AIMS Mathematics, 6(2021), 8390-8404.
J. Zuo, T. An and M. Li, Superlinear Kirchhoff-type problems of the fractional p-Laplacian
without the (AR) condition, Boundary Value Problems, 2018, 1-13.