Abstract
We deal with a nonlinear elliptic weighted system of Lane–Emden type in \(\mathbb R^N\), \(N \ge 3\), by exploiting its equivalence with a fourth-order quasilinear elliptic equation involving a suitable “sublinear” term. By overcoming the loss of compactness of the problem with some compact imbeddings in weighted \(L^p\)-spaces, we establish existence and multiplicity results by means of a generalized Weierstrass Theorem and a variant of the Symmetric Mountain Pass Theorem stated by R. Kajikiya for subquadratic functionals. These results, which generalize previous ones stated by the same authors, apply in particular to a biharmonic equation under Navier conditions in \(\mathbb R^N\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Angenent, S., Van der Vorst, R.: A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology. Math. Z. 231(2), 203–248 (1999)
Barile, S., Salvatore, A.: Weighted elliptic systems of Lane–Emden type in unbounded domains. Mediterr. J. Math. 9(3), 409–422 (2012)
Barile, S., Salvatore, A.: Some multiplicity and regularity results for perturbed elliptic systems. Dyn. Syst. Appl. 6, 58–64 (2012)
Barile, S., Salvatore, A.: Existence and multiplicity results for some elliptic systems in unbounded cylinders. Milan J. Math. 81(1), 99–120 (2013)
Barile, S., Salvatore, A.: Existence and multiplicity results for some Lane—Emden elliptic systems: subquadratic case. Adv. Nonlinear Anal. 4(1), 25–35 (2015)
Barile, S., Salvatore, A.: Some results on weighted subquadratic Lane–Emden elliptic systems in unbounded domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27(1), 89–103 (2016)
Bernis, F., García Azorero, J., Peral, I.: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 1(2), 219–240 (1996)
Bonheure, D., Dos Santos, E.M., Ramos, M.: Ground state and non-ground state solutions of some strongly coupled elliptic systems. Trans. Am. Math. Soc. 364(1), 447–491 (2012)
Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York (2011)
Candela, A.M., Salvatore, A.: Elliptic systems in unbounded domains. Complex Var. Elliptic Equ. 56(12), 1143–1153 (2011)
Clément, Ph, De Figueiredo, D.G., Mitidieri, E.: Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 17(5–6), 923–940 (1992)
Clément, Ph, Mitidieri, E.: On a class of nonlinear elliptic systems. Nonlinear Evol. Equ. Appl. Res. Inst. Math. Sci. Kyoto 1997, 132–140 (1009)
De Figueiredo, D.G., Felmer, P.: On superquadratic elliptic systems. Trans. Am. Math. Soc. 343(1), 99–116 (1994)
De Figueiredo, D.G., Ruf, B.: Elliptic systems with nonlinearities of arbitrary growth. Mediterr. J. Math. 1(4), 417–431 (2004)
Dos Santos, E.M.: Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation. J. Math. Anal. Appl. 342(1), 277–297 (2008)
Felmer, P., Martínez, S.: Existence and uniqueness of positive solutions to certain differential systems. Adv. Differ. Equ. 3(4), 575–593 (1998)
Gazzola, F., Grunau, H.C., Sweers, G.: Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Mathematics, vol. 1991. Springer, Berlin (2010)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Heidelberg (1983)
Hulshof, J., Van der Vorst, R.: Differential systems with strongly indefinite variational structure. J. Funct. Anal. 114(1), 32–58 (1993)
Kajikiya, R.: A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J. Funct. Anal. 225(2), 352–370 (2005)
Lv, Y.: Existence and multiplicity of solutions for a class of sublinear Schrödinger-Maxwell equations. Bound. Value Probl. 177, 1–22 (2013)
Mitidieri, E.: A Rellich type identity and applications. Commun. Partial Differ. Equ. 18(1–2), 125–151 (1993)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986)
Salvatore, A.: Multiple solutions for elliptic systems with nonlinearities of arbitrary growth. J. Differ. Equ. 244(10), 2529–2544 (2008)
Salvatore, A.: Infinitely many solutions for symmetric and non-symmetric elliptic systems. J. Math. Anal. Appl. 366(2), 506–515 (2010)
Van der Vorst, R.: Variational identities and applications to differential systems. Arch. Ration. Mech. Anal. 116(4), 375–398 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are partially supported by INDAM-GNAMPA Project 2015 “Metodi variazionali e topologici applicati allo studio di problemi ellittici non lineari”.
Rights and permissions
About this article
Cite this article
Barile, S., Salvatore, A. Some New Results on Subquadratic Lane–Emden Elliptic Systems. Mediterr. J. Math. 14, 31 (2017). https://doi.org/10.1007/s00009-016-0818-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-016-0818-1