Solutions of  $p$-Kirchhoff type problems with critical nonlinearity in $\mathbb{R}^N$

Song, Yueqiang; Shi, Shaoyun

doi:10.22436/jnsa.011.02.01

Geodesic

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Solutions of $p$-Kirchhoff type problems with critical nonlinearity in $\mathbb{R}^N$

Song, Yueqiang ¹ ; Shi, Shaoyun ²

¹ Scientific Research Department, Changchun Normal University, Changchun 130032, Jilin, P. R. China
² School of Mathematics & State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, Jilin, P. R. China

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 2, p. 172-188.

Voir la notice de l'article provenant de la source International Scientific Research Publications

Résumé

In this paper, we are interested in the existence of weak solutions for the fractional p-Laplacian equation with critical nonlinearity in $\mathbb R^N$. By using fractional version of concentration compactness principle together with variational method, we obtain the existence and multiplicity of solutions for the above problem.

DOI : 10.22436/jnsa.011.02.01

Classification : 35J60, 47G20
Keywords: Fractional $p$-Laplacian equation, critical nonlinearity, variational method, critical points

Affiliations des auteurs :

Song, Yueqiang ¹ ; Shi, Shaoyun ²

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     title = {Solutions of  {\(p\)-Kirchhoff} type problems with critical nonlinearity in {\(\mathbb{R}^N\)}},
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     pages = {172-188},
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Song, Yueqiang ; Shi, Shaoyun . Solutions of  \(p\)-Kirchhoff type problems with critical nonlinearity in \(\mathbb{R}^N\). Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 2, p. 172-188. doi : 10.22436/jnsa.011.02.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.02.01/

Bibliographie
Cité par

[1] Applebaum, D. Lévy processes–from probability to finance and quantum groups , Notices Amer. Math. Soc., Volume 51 (2004), pp. 1336-1347 | Zbl

[2] Autuori, G.; Fiscella, A.; P. Pucci Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity , Nonlinear Anal., Volume 125 (2015), pp. 699-714 | DOI | Zbl

[3] Autuori, G.; P. Pucci Elliptic problems involving the fractional Laplacian in $R^N$ , J. Differential Equations, Volume 255 (2013), pp. 2340-2362 | DOI

[4] Barrios, B.; Colorado, E.; Pablo, A. de; U. Sánchez On some critical problems for the fractional Laplacian operator, J. Differential Equations, Volume 252 (2012), pp. 6133-6162 | DOI

[5] V. Benci On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., Volume 274 (1982), pp. 533-572 | DOI | Zbl

[6] Brézis; Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math., Volume 34 (1983), pp. 437-477 | DOI

[7] Caffarelli, L.; diffusions, Non-local; drifts and games Nonlinear partial differential equations, Abel Symp. , Springer, Heidelberg, Volume 7 (2012), pp. 37-52

[8] X.-J. Chang Ground state solutions of asymptotically linear fractional Schrödinger equations , J. Math. Phys., Volume 54 (2013), pp. 1-10 | Zbl | DOI

[9] M. de Souza On a class of nonhomogeneous fractional quasilinear equations in $R^n$ with exponential growth , NoDEA Nonlinear Differential Equations Appl., Volume 22 (2015), pp. 499-511 | DOI | Zbl

[10] Nezza, E. Di; Palatucci, G.; E. Valdinoci Hitchhiker’s guide to the fractional Sobolev spaces , Bull. Sci. Math., Volume 136 (2012), pp. 521-573 | DOI | Zbl

[11] Ding, Y.-H.; Lin, F.-H. Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, Volume 30 (2007), pp. 231-249 | DOI

[12] Dong, W.; Xu, J.-F.; Z.-L. Wei Infinitely many weak solutions for a fractional Schrödinger equation, Bound. Value Probl., Volume 2014 (2014), pp. 1-14 | DOI | Zbl

[13] Gou, T.-X.; Sun, H.-R. Solutions of nonlinear Schrödinger equation with fractional Laplacian without the Ambrosetti- Rabinowitz condition, Appl. Math. Comput., Volume 257 (2015), pp. 409-416 | Zbl | DOI

[14] Iannizzotto, A.; Liu, S.-B.; Perera, K.; M. Squassina Existence results for fractional p-Laplacian problems via Morse theory , Adv. Calc. Var., Volume 9 (2016), pp. 101-125 | DOI | Zbl

[15] N. Laskin Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, Volume 268 (2000), pp. 298-305 | DOI

[16] N. Laskin Fractional Schrödinger equation , Phys. Rev. E, Volume 66 (2002), pp. 1-7 | DOI

[17] Liang, S.-H.; Shi, S.-Y. Soliton solutions to Kirchhoff type problems involving the critical growth in $R^N$, Nonlinear Anal., Volume 81 (2013), pp. 31-41 | DOI | Zbl

[18] Liang, S.-H.; Zhang, J.-H. Existence of solutions for Kirchhoff type problems with critical nonlinearity in $R^3$ , Nonlinear Anal. Real World Appl., Volume 17 (2014), pp. 126-136 | DOI

[19] Liang, S.-H.; J.-H. Zhang Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in $R^N$, J. Math. Phys., Volume 57 (2016), pp. 1-13 | Zbl | DOI

[20] Liang, S.-H.; J.-H. Zhang Multiplicity of solutions for the noncooperative Schrödinger-Kirchhoff system involving the fractional p-Laplacian in $R^N$ , Z. Angew. Math. Phys., Volume 68 (2017), pp. 1-18 | Zbl | DOI

[21] P.-L. Lions The concentration-compactness principle in the calculus of variations, The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 1 (1984), pp. 223-283 | DOI | Zbl

[22] Liu, J.; Liao, J.-F.; C.-L. Tang Positive solutions for Kirchhoff-type equations with critical exponent in $R^N$, J. Math. Anal. Appl., Volume 429 (2015), pp. 1153-1172 | DOI

[23] Bisci, G. Molica; V. D. Rădulescu Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, Volume 54 (2015), pp. 2985-3008 | Zbl | DOI

[24] Bisci, G. Molica; Rădulescu, V. D. Multiplicity results for elliptic fractional equations with subcritical term , NoDEA Nonlinear Differential Equations Appl., Volume 22 (2015), pp. 721-739 | DOI | Zbl

[25] Bisci, G. Molica; D. Repovš Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., Volume 420 (2014), pp. 167-176 | DOI

[26] A. Ourraoui On a p-Kirchhoff problem involving a critical nonlinearity , C. R. Math. Acad. Sci. Paris, Volume 352 (2014), pp. 295-298 | DOI | Zbl

[27] Pucci, P.; S. Saldi Critical stationary Kirchhoff equations in $R^N$ involving nonlocal operators , Rev. Mat. Iberoam., Volume 32 (2016), pp. 1-22 | DOI | Zbl

[28] Pucci, P.; Xiang, M.-Q.; Zhang, B.-L. Multiple solutions for nonhomogeneous Schrödinger -Kirchhoff type equations involving the fractional p-Laplacian in $R^N$, Calc. Var. Partial Differential Equations, Volume 54 (2015), pp. 2785-2806 | DOI | Zbl

[29] Pucci, P.; Xiang, M.-Q.; B.-L. Zhang Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal., Volume 5 (2016), pp. 27-55 | DOI | Zbl

[30] P. H. Rabinowitz Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986 | DOI

[31] Ros-Oton, X.; Serra, J. Nonexistence results for nonlocal equations with critical and supercritical nonlinearities , Comm. Partial Differential Equations, Volume 40 (2015), pp. 115-133 | Zbl | DOI

[32] Servadei, R.; Valdinoci, E. Fractional Laplacian equations with critical Sobolev exponent, Rev. Mat. Complut., Volume 28 (2015), pp. 655-676 | DOI

[33] Servadei, R.; E. Valdinoci The Brezis-Nirenberg result for the fractional Laplacian , Trans. Amer. Math. Soc., Volume 367 (2015), pp. 67-102 | DOI

[34] Shang, X.-D.; Zhang, J.-H. Concentrating solutions of nonlinear fractional Schrdinger equation with potentials, J. Differential Equations, Volume 258 (2015), pp. 1106-1128 | DOI

[35] K.-M. Teng Multiple solutions for a class of fractional Schrdinger equations in $R^N$, Nonlinear Anal. Real World Appl., Volume 21 (2015), pp. 76-86 | DOI

[36] C. Torres On superlinear fractional p-Laplacian in $R^n$, ArXiv, Volume 2014 (2014), pp. 1-12

[37] Willem, M. Minimax theorems , Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 1996

[38] Xiang, M.-Q.; Zhang, B.-L.; M. Ferrara Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian , J. Math. Anal. Appl., Volume 424 (2015), pp. 1021-1041 | DOI | Zbl

[39] Xiang, M.-Q.; Zhang, B.-L.; M. Ferrara Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities, Proc. A, Volume 471 (2015), pp. 1-14 | Zbl | DOI

[40] Xiang, M.-Q.; Zhang, B.-L.; V. D. Rădulescu Existence of solutions for perturbed fractional p-Laplacian equations , J. Differential Equations, Volume 260 (2016), pp. 1392-1413 | Zbl | DOI

[41] Xiang, M.-Q.; Zhang, B.-L.; X. Zhang A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in $R^N$, Adv. Nonlinear Stud., Volume 17 (2017), pp. 611-640 | DOI | Zbl

[42] Zhang, X.; Zhang, B.-L.; M.-Q. Xiang Ground states for fractional Schrödinger equations involving a critical nonlinearity , Adv. Nonlinear Anal., Volume 5 (2016), pp. 293-314 | DOI | Zbl

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