Abstract
The main purpose of this paper is to investigate the existence of nontrivial solutions to a class of quasilinear non-local problems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth (subcritical polynomial growth) at \(\infty \). Some existence results for nontrivial solution are obtained using mountain pass theorem combined with the fractional Moser–Trudinger inequality.
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References
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Brasco, L., Parini, E.: The second eigenvalue of the fractional \(p\)-Laplacian (preprint)
Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana Series, vol. 20. Springer (2016)
Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994)
Costa, D.G., Miyagaki, O.H.: Nontrivial solutions for perturbations of the \(p\)-Laplacian on unbounded domains. J. Math. Anal. Appl. 193, 737–755 (1995)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
de Figueiredo, D.G., doÓ, J.M., Ruf, B.: Elliptic equations in \({\mathbb{R}}^2\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3, 139–153 (1995)
doÓ, J.M.: Semilinear Dirichlet problems for the \(N\)-Laplacian in \({\mathbb{R}}^{N}\) with nonlinearities in the critical growth range. Differ. Integral. Equ. 9, 967–979 (1996)
Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5, 373–386 (2014)
Fiscella, A., Molica Bisci, G., Servadei, R.: Bifurcation and multiplicity results for critical nonlocal fractional problems. Bull. Sci. Math 140, 14–35 (2016)
Brezis, H., Nirenberg, L.: positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)
Iannizzotto, A., Liu, S.B., Perera, K., Squassina, M.: Existence results for fractional \(p\)-Laplacian problems via Morse theory. Adv. Calc. Var. (2014). https://doi.org/10.1515/acv-2014-0024
Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. arXiv:1411.2956 (preprint)
Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional \(p\)-eigenvalue problems. Asymptot. Anal. 88, 233–245 (2014)
Iannizzotto, A., Squassina, M.: \(1/2\)-Laplacian problems with exponential nonlinearity. J. Math. Anal. Appl. 414, 372–385 (2014)
Jeanjean, L.: On the existence of bounded Palais–Smale sequences and applications to a Landesman–Lazer-type problem set on \({\mathbb{R}}^N\). Proc. R. Soc. Edinb. 129, 787–809 (1999)
Lam, N., Lu, Guozhen: N-Laplacian equations in \({\mathbb{R}}^N\) with subcritical and critical growth without the Ambrosetti–Rabinowitz condition. Adv. Nonlinear Stud. 13, 289–308 (2013)
Liu, Z.L., Wang, Z.Q.: On the Ambrosetti–Rabinowitz superlinear condition. Adv. Nonlinear Stud. 4, 563–574 (2004)
Li, G.B., Zhou, H.S.: Asymptotically linear Dirichlet problem for the \(p\)-Laplacian. Nonlinear Anal. 43, 1043–1055 (2001)
Liu, S.B., Li, S.J.: Infinitely many solutions for a superlinear elliptic equation. Acta. Math. Sin. 46, 625–630 (2003)
Martinazzi, L.: Fractional Adams–Moser–Trudinger type inequalities. arXiv:1506.00489 (preprint)
Molica Bisci, G.: Sequence of weak solutions for fractional equations. Math. Res. Lett. 21, 241–253 (2014)
Molica Bisci, G., Rǎdulescu, D.V.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial Differ. Equ. 54, 2985–3008 (2015)
Molica Bisci, G., Rǎdulescu, D.V., Servadei, R.: Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge University Press, Cambridge (2016)
Parini, E., Ruf, B.: On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces (preprint)
Perera, K., Agarwal, R.P., O’Regan, D.: Morse theoretic aspects of \(p\)-Laplacian operators. Mathematical Surveys and Monographs, vol. 161. American Mathematical Society, Providence (2010)
Perera, K., Squassina, M., Yang, Y.: Bifurcation and multiplicity results for critical fractional \(p\)-Laplacian problems. arXiv:1407.8061 (preprint)
Schechter, M., Zou, W.M.: Superlinear problems. Pac. J. Math. 214, 145–160 (2004)
Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)
Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)
Xiang, M.Q., Molica Bisci, G., Tian, G.H., Zhang, B.L.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian. Nonlinearity 29, 357–374 (2015)
Zhang, B.L., Ferrara, M.: Multiplicity of soutions for a class of superlinear non-local fractional equations. Complex Var. Elliptic Equ. 60, 583–595 (2015)
Zhang, B.L., Molica Bisci, G., Servadei, R.: Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28, 2247–2264 (2015)
Zhang, Y.M., Shen, Y.T.: Existence of solutions for elliptic equations without superquadraticity condition. Front. Math. China 7, 587–595 (2012)
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This research is supported by the NSFC (Nos. 11661070, 11764035 and 11571176).
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Pei, R. Fractional p-Laplacian Equations with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition. Mediterr. J. Math. 15, 66 (2018). https://doi.org/10.1007/s00009-018-1115-y
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DOI: https://doi.org/10.1007/s00009-018-1115-y