| Citation: |
J. Vanterler da C. Sousa, Gabriela L. Araújo, Maria V. S. Sousa, Amália R. E. Pereira. MULTIPLICITY OF SOLUTIONS FOR FRACTIONAL κ(X)-LAPLACIAN EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1543-1578. doi: 10.11948/20230293
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MULTIPLICITY OF SOLUTIONS FOR FRACTIONAL κ(X)-LAPLACIAN EQUATIONS
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Abstract
In this present paper, we first discuss some results of the energy functional on the Nehari manifold. Furthermore, we are interested in a compactness result and in estimates involving minimax levels over the $\psi$-fractional space $\mathcal{H}^{\alpha, \beta; \psi}_ { \kappa(\xi)}(\Omega)$. In this sense, the condition of Palais-Smale is discussed. In other words, we are concerned with the multiplicity of solutions to a class of quasilinear fractional problems with super-linear growth involving variable exponents through the previously discussed results, in particular via the Lions concentration-compaction principle.
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References
[1] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 2002, 213–259. [2] G. Alessandrini, Critical points of solutions to the $p$-Laplace equation in dimension two, Bollettino della Unione Matematica Italiana, Sezione A., 1987, 1, 239–246. [3] C. O. Alves, Existence of solution for a degenerate $p(x)$-Laplacian equation in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 2008, 345, 731–742. doi: 10.1016/j.jmaa.2008.04.060 CrossRef $p(x)$-Laplacian equation in
$\mathbb{R}^{N}$ [4] C. O. Alves and J. L. P. Barreiro, Existence and multiplicity of solutions for a $p(x)$-Laplacian equation with critical growth, J. Math. Anal. Appl., 2013, 403, 143–154. doi: 10.1016/j.jmaa.2013.02.025 CrossRef $p(x)$-Laplacian equation with critical growth" target="_blank">Google Scholar
[5] C. O. Alves and J. L. P. Barreiro, Multiple solutions for a class of quasilinear problems involving variable exponents, Asymptotic Anal., 2016, 96(2), 161–184. doi: 10.3233/ASY-151340 [6] C. O. Alves and M. C. Ferreira, Existence of solutions for a class of $p(x)$-Laplacian equations involving a concave-convex nonlinearity with critical growth in $\mathbb{R}^{N}$, Topol. Methods Nonlinear Anal., 2015, 45(2), 399–422. doi: 10.12775/TMNA.2015.020 CrossRef $p(x)$-Laplacian equations involving a concave-convex nonlinearity with critical growth in
$\mathbb{R}^{N}$ [7] A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 1994, 122, 519–543. doi: 10.1006/jfan.1994.1078 [8] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 1973, 14, 349–381. doi: 10.1016/0022-1236(73)90051-7 [9] S. N. Antontseva and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions, Nonlinear Anal., 2005, 60, 515–545. doi: 10.1016/j.na.2004.09.026 [10] E. Azroul and A. Benkirane, On a nonlocal problem involving the fractional $p(x, \cdot)$-Laplacian satisfying Cerami condition, Disc. Cont. Dyn. Sys.-S. S, 2021, 14(10). [11] E. D. Benedetto, $C^{1, \alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 1983, 7, 827–850. doi: 10.1016/0362-546X(83)90061-5 CrossRef $C^{1, \alpha}$ local regularity of weak solutions of degenerate elliptic equations" target="_blank">Google Scholar
[12] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equ., 2003, 193, 481–499. doi: 10.1016/S0022-0396(03)00121-9 [13] D. M. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^{N}$, Ann. Inst. H. Poincare Anal. Non Lineaire, 1996, 13(5), 567–588. doi: 10.1016/s0294-1449(16)30115-9 [14] J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific, 1999. [15] R. Chammem, A. Ghanmi and A. Sahbani, Existence of solution for a singular fractional Laplacian problem with variable exponents and indefinite weights, Complex Varia. Ellip. Equ., 2021, 66(8), 1320–1332. doi: 10.1080/17476933.2020.1756270 [16] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 2006, 66(4), 1383–1406. doi: 10.1137/050624522 [17] L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, 2011. [18] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 1974, 47, 324–353. doi: 10.1016/0022-247X(74)90025-0 [19] R. Ezati and N. Nyamoradi, Existence and multiplicity of solutions to a $\psi$-Hilfer fractional $p$-Laplacian equations, Asian-European J. Math., 2022, 2350045. $\psi$-Hilfer fractional
$p$ [20] R. Ezati and N. Nyamoradi, Existence of solutions to a Kirchhoff $\psi$-Hilfer fractional $p$‐Laplacian equations, Math. Meth. Appl. Sci., 2021, 44(17), 12909–12920. doi: 10.1002/mma.7593 CrossRef $\psi$-Hilfer fractional
$p$ [21] X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 2001, 262(2), 749–760. doi: 10.1006/jmaa.2001.7618 CrossRef $W^{k, p(x)}(\Omega)$" target="_blank">Google Scholar
[22] X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 2005, 302(2), 306–317. doi: 10.1016/j.jmaa.2003.11.020 CrossRef $p(x)$-Laplacian Dirichlet problem" target="_blank">Google Scholar
[23] X. Fan, Y. Zhao and D. Zhao, Compact embedding theorems with symmetry of Strauss-Lions type for the space $W^{1, p(x)}(\mathbb{R}^{N})$, J. Math. Anal. Appl., 2001, 255, 333–348. doi: 10.1006/jmaa.2000.7266 CrossRef $W^{1, p(x)}(\mathbb{R}^{N})$" target="_blank">Google Scholar
[24] X.-L. Fan and Q.-H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Analysis: Theory, Methods & Applications, 2003, 52(8), 1843–1852. $p(x)$-Laplacian Dirichlet problem" target="_blank">Google Scholar
[25] Y. Fu, The principle of concentration compactness in $L~{p(x)}$ spaces and its application, Nonlinear Anal., 2009, 71, 1876–1892. doi: 10.1016/j.na.2009.01.023 CrossRef $L~{p(x)}$ spaces and its application" target="_blank">Google Scholar
[26] Y. Fu and X. Zhang, Multiple solutions for a class of $p(x)$-Laplacian equations in involving the critical exponent, Proceedings of the Royal Society A: Math. Phys. Engine. Sci., 2010, 466(2118), 1667–1686. $p(x)$-Laplacian equations in involving the critical exponent" target="_blank">Google Scholar
[27] J. P. Garcia Azorero and I. P. Alonsl, Existence and non-uniqueness for the $p$-Lapacian: non-linear eigenvalues, Commun. Partial Differ. Equ., 1987, 12(12), 1389–1430. $p$-Lapacian: non-linear eigenvalues" target="_blank">Google Scholar
[28] M. R. Hamidi and N. Nyamoradi, On boundary value problem for fractional differential equations, Bull. Iranian Math. Soc., 2017, 43(3), 789–805. [29] Q. Han and F. Ling, Elliptic Partial Differential Equations (Courant Lecture Notes), New York, 1997. [30] Z. He and L. Miao, Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space, AIMS Math., 2021, 6(6), 6171–6179. doi: 10.3934/math.2021362 [31] T. S. Hsu, H. L. Lin and C. C. Hu, Multiple positive solutions of quasilinear elliptic equations in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 2012, 388, 500–512. doi: 10.1016/j.jmaa.2011.11.010 [32] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire., 1984, 1, 223–283. doi: 10.1016/s0294-1449(16)30422-x [33] O. H. Miyagaki, D. Motreanu and F. R. Pereira, Multiple solutions for a fractional elliptic problem with critical growth, J. Diff. Equ., 2020, 269(6), 5542–5572. doi: 10.1016/j.jde.2020.04.010 [34] P. Montecchiari, Multiplicity results for a class of semilinear elliptic equations on $\mathbb{R}^{N}$, Rend. Sem. Mat. Univ. Padova., 1996, 95, 217–252. [35] N. Nyamoradi, The Nehari manifold and its application to a fractional boundary value problem, Diff. Equ. Dyn. Sys., 2013, 21, 323–340. doi: 10.1007/s12591-013-0175-5 [36] N. Nyamoradi, Existence and multiplicity of solutions to a singular elliptic system with critical Sobolev–Hardy exponents and concave–convex nonlinearities, J. Math. Anal. Appl., 2012, 396(1), 280–293. doi: 10.1016/j.jmaa.2012.06.026 [37] W. Orlicz, Uber konjugierte exponentenfolgen, Studia Math., 1931, 3, 200–211. doi: 10.4064/sm-3-1-200-211 [38] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Vol. 1748 of Lecture Notes in Math. Springer-Verlag, 2000. [39] X. Shang and Z. Wang, Existence of solutions for discontinuous $p(x)$-Laplacian problems with critical exponents, Electron. J. Diff. Equ., 2012, 2012(25), 1–12. $p(x)$-Laplacian problems with critical exponents" target="_blank">Google Scholar
[40] H. M. Srivastava and J. Vanterler da C. Sousa, Multiplicity of solutions for fractional-order differential equations via the $\kappa(x)$-Laplacian operator and the genus theory, Fractal Frac., 2022, 6(9), 481. doi: 10.3390/fractalfract6090481 CrossRef $\kappa(x)$-Laplacian operator and the genus theory" target="_blank">Google Scholar
[41] J. Vanterler da C. Sousa, Nehari manifold and bifurcation for a $\psi$-Hilfer fractional $p$‐Laplacian, Math. Meth. Appl. Sci., 2021. DOI: 10.1002/mma.7296. CrossRef $\psi$-Hilfer fractional
$p$ [42] J. Vanterler da C. Sousa, Existence and uniqueness of solutions for the fractional differential equations with $p$-Laplacian in $\mathcal{H}^{\nu, \mu; \psi}_{p}$, J. Appl. Anal. Comput., 2022, 12(2), 622–661. $p$-Laplacian in
$\mathcal{H}^{\nu, \mu; \psi}_{p}$ [43] J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 2018, 60, 7291. $\psi$-Hilfer fractional derivative" target="_blank">Google Scholar
[44] J. Vanterler da C. Sousa, C. T. Ledesma, M. Pigossi and J. Zuo, Nehari Manifold for Weighted Singular Fractional $p$-Laplace Equations, Bull. Braz. Math. Soc., 2022, 1–31. [45] J. Vanterler da C. Sousa, L. S. Tavares and C. E. T. Ledesma, A variational approach for a problem involving a $\psi$-Hilfer fractional operator, J. Appl. Anal. Comput., 2021, 11(3), 1610–1630. $\psi$-Hilfer fractional operator" target="_blank">Google Scholar
[46] J. Vanterler da C. Sousa, J. Zuo and D. O'Regan, The Nehari manifold for a $\psi$-Hilfer fractional $p$-Laplacian, Applicable Anal., 2021, 1–31. $\psi$-Hilfer fractional
$p$ [47] H. Yin and Z. Yang, Existence of multiple solutions for quasilinear elliptic equations in $\mathbb{R}^{N}$, Electr. J. Diff. Equ., 2014, 2014(17), 1–22. [48] C. Zhang and X. Zhang, Renormalized solutions for the fractional $p(x)$-Laplacian equation with $L^{1}$ data, Nonlinear Anal., 2020, 190, 111610. doi: 10.1016/j.na.2019.111610 CrossRef $p(x)$-Laplacian equation with
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J. Vanterler da C. Sousa, Gabriela L. Araújo, Maria V. S. Sousa, Amália R. E. Pereira. MULTIPLICITY OF SOLUTIONS FOR FRACTIONAL κ(X)-LAPLACIAN EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1543-1578. doi: 10.11948/20230293
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