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Solutions for fourth order elliptic equations on $$\mathbb{R}^N$$ involving $$u \Delta(u^{2})$$ and sign-changing potentials. (English) Zbl 1418.35128
Summary: We obtain existence and multiplicity results for fourth order elliptic equations on $$\mathbb{R}^N$$ involving $$u \Delta(u^2)$$ and sign-changing potentials. Our results generalize some recent results on this kind of problems. To study this kind of problems, we first consider the case that the potential $$V$$ is coercive so that the working space can be compactly embedded into Lebesgue spaces. Then we studied the case that the potential $$V$$ is bounded so that the working space is exactly $$H^2(\mathbb{R}^N)$$, which can not be compactly embedded into Lebesgue spaces anymore. To deal with this more difficult case, we study the weak continuity of the term in the energy functional corresponding to the term $$u \Delta(u^2)$$ in the equation.
##### MSC:
 35J30 Higher-order elliptic equations 35J62 Quasilinear elliptic equations
##### Keywords:
biharmonic operator; quasilinear equation
Full Text:
##### References:
 [1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381, (1973) · Zbl 0273.49063 [2] Bartsch, T.; Li, S., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28, 419-441, (1997) · Zbl 0872.58018 [3] Bartsch, T.; Liu, Z.; Weth, T., Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations, 29, 25-42, (2004) · Zbl 1140.35410 [4] Bartsch, T.; Wang, Z. Q., Existence and multiplicity results for some superlinear elliptic problems on $$\mathbf{R}^N$$, Comm. Partial Differential Equations, 20, 1725-1741, (1995) · Zbl 0837.35043 [5] Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 486-490, (1983) · Zbl 0526.46037 [6] Chabrowski, J.; do Ó, J. Marcos, On some fourth-order semilinear elliptic problems in $$\mathbb{R}^N$$, Nonlinear Anal., 49, 861-884, (2002) · Zbl 1011.35045 [7] Chang, K.-c., Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications, vol. 6, (1993), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA [8] Che, G.; Chen, H., Existence of multiple nontrivial solutions for a class of quasilinear Schrödinger equations on $$\mathbb{R}^N$$, Bull. Belg. Math. Soc. Simon Stevin, 25, 39-53, (2018) · Zbl 1394.35191 [9] Chen, H.; Liu, S., Standing waves with large frequency for 4-superlinear Schrödinger-Poisson systems, Ann. Mat. Pura Appl. (4), 194, 43-53, (2015) · Zbl 1309.35008 [10] Chen, S.; Liu, J.; Wu, X., Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on $$\mathbb{R}^N$$, Appl. Math. Comput., 248, 593-601, (2014) · Zbl 1338.35152 [11] Cheng, B.; Tang, X., High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential, Comput. Math. Appl., 73, 27-36, (2017) · Zbl 1368.35111 [12] Colin, M.; Jeanjean, L., Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56, 213-226, (2004) · Zbl 1035.35038 [13] Ding, Y., Variational Methods for Strongly Indefinite Problems, Interdisciplinary Mathematical Sciences, vol. 7, (2007), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1133.49001 [14] Kajikiya, R., A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225, 352-370, (2005) · Zbl 1081.49002 [15] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3, 441-472, (1998) · Zbl 0947.35061 [16] Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32, 537-578, (1990) · Zbl 0725.73057 [17] Li, S. J.; Willem, M., Applications of local linking to critical point theory, J. Math. Anal. Appl., 189, 6-32, (1995) · Zbl 0820.58012 [18] Liu, J.; Su, J., Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258, 209-222, (2001) · Zbl 1050.35025 [19] Liu, J. Q., The Morse index of a saddle point, J. Systems Sci. Math. Sci., 2, 32-39, (1989) · Zbl 0732.58011 [20] Liu, J. Q.; Li, S. J., An existence theorem for multiple critical points and its application, Kexue Tongbao (Chinese), 29, 1025-1027, (1984) [21] Liu, J.-q.; Wang, Y.-q.; Wang, Z.-Q., Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187, 473-493, (2003) · Zbl 1229.35268 [22] Liu, S., On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45, 1-9, (2012) · Zbl 1247.35149 [23] Liu, S.; Wu, Y., Standing waves for 4-superlinear Schrödinger-Poisson systems with indefinite potentials, Bull. Lond. Math. Soc., 49, 226-234, (2017) · Zbl 1379.35081 [24] Liu, S.; Zhou, J., Standing waves for quasilinear Schrödinger equations with indefinite potentials, J. Differential Equations, 265, 3970-3987, (2018) · Zbl 1404.35130 [25] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, vol. 74, (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0676.58017 [26] Rabinowitz, P. H., Minimax methods and their application to partial differential equations, (Seminar on Nonlinear Partial Differential Equations. Seminar on Nonlinear Partial Differential Equations, Berkeley, Calif., 1983. Seminar on Nonlinear Partial Differential Equations. Seminar on Nonlinear Partial Differential Equations, Berkeley, Calif., 1983, Math. Sci. Res. Inst. Publ., vol. 2, (1984), Springer: Springer New York), 307-320 [27] Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, (CBMS Regional Conference Series in Mathematics, vol. 65, (1986), Conference Board of the Mathematical Sciences: Conference Board of the Mathematical Sciences Washington, DC) · Zbl 0609.58002 [28] Wang, W.; Zang, A.; Zhao, P., Multiplicity of solutions for a class of fourth elliptic equations, Nonlinear Anal., 70, 4377-4385, (2009) · Zbl 1162.35355 [29] Yang, Y.; Zhang, J., Existence of solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. Appl., 351, 128-137, (2009) · Zbl 1179.35140 [30] Ye, Y.; Tang, C.-L., Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 394, 841-854, (2012) · Zbl 1248.35069 [31] Yin, Y.; Wu, X., High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375, 699-705, (2011) · Zbl 1206.35120 [32] Zhang, J.; Li, S., Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60, 221-230, (2005) · Zbl 1103.35027 [33] Zhang, W.; Tang, X.; Zhang, J., Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407, 359-368, (2013) · Zbl 1311.35095 [34] Zhang, W.; Tang, X.; Zhang, J., Ground states for a class of asymptotically linear fourth-order elliptic equations, Appl. Anal., 94, 2168-2174, (2015) · Zbl 1326.35119 [35] Zhou, J.; Wu, X., Sign-changing solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. Appl., 342, 542-558, (2008) · Zbl 1138.35335
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