Abstract
We establish the existence of two opposite constant sign solutions for a general noncoercive quasilinear elliptic system with homogeneous Dirichlet boundary conditions. In the case where the system has a variational structure, by strengthening the hypotheses, we obtain a third nontrivial solution which is sign changing in the sense that one cannot have both components of the new solution of the same constant sign. Our approach relies on a suitable method of sub-supersolutions combined with truncation and variational arguments that does not require a subcritical growth condition.
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Anane, A.: Simplicité et isolation de la première valeur propre du p-Laplacien avec poids. C. R. Acad. Sci., Sér. I Math. (Paris) 305, 725–728 (1987)
Averna, D., Marano, S.A., Motreanu, D.: Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity. Bull. Aust. Math. Soc. 77, 285–303 (2008)
Bartsch, T., Liu, Z.: Multiple sign-changing solutions of a quasilinear elliptic problem involving the p-Laplacian, Commun. Contemp. Math. 6, 245–258 (2004)
Carl, S., Grossmann, C.: Monotone enclosure for elliptic and parabolic systems with nonmonotone nonlinearities. J. Math. Anal. Appl. 151, 190–202 (1990)
Carl, S., Jerome, J.W.: Trapping region for discontinuous quasilinear elliptic systems of mixed monotone type. Nonlinear Anal. 51, 843–863 (2002)
Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and their Inequalities. Comparison Principles and Applications. Springer, New York (2007)
Carl, S., Motreanu, D.: Constant-sign and sign-changing solutions for nonlinear eigenvalue problems. Nonlinear Anal. 68, 2668–2676 (2008)
Carl, S., Naniewicz, Z.: Vector quasi-hemivariational inequalities and discontinuous elliptic systems. J. Glob. Optim. 34, 609–634 (2006)
Carl, S., Perera, K.: Sign-changing and multiple solutions for the p-Laplacian. Abstr. Appl. Anal. 7, 613–625 (2002)
Cuesta, M., de Figueiredo, D., Gossez, J.-P.: The beginning of the Fučik spectrum for the p-Laplacian. J. Differ. Equ. 159, 212–238 (1999)
Gasiński, L., Papageorgiou, N.S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman & Hall/CRC, Boca Raton, FL (2005)
Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton, FL (2006)
Koizumi, E., Schmidt, K.: Ambrosetti-Prodi-type problems for quasilinear elliptic problems. Differ. Integral Equ. 18, 241–262 (2005)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: A unified approach for constant sign and nodal solutions. Adv. Differ. Equ. 12, 1363–1392 (2007)
Motreanu, D., Perera, K.: Nontrivial solutions of Neumann p-Laplacian systems. Topol. Methods Nonlinear Anal. 34, 41–48 (2009)
Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)
Zhang, Z., Chen, J., Li, S.: Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian. J. Differ. Equ. 201, 287–303 (2004)
Zhang, J., Zhang, Z.: Existence results for some nonlinear elliptic systems. Nonlinear Anal. 71, 2840–2846 (2009)
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Supported by National Natural Science Foundation of China (10671195, 10831005, 10971046).
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Motreanu, D., Zhang, Z. Constant Sign and Sign Changing Solutions for Systems of Quasilinear Elliptic Equations. Set-Valued Anal 19, 255–269 (2011). https://doi.org/10.1007/s11228-010-0142-z
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DOI: https://doi.org/10.1007/s11228-010-0142-z
Keywords
- Quasilinear elliptic system
- Sub-supersolution
- Comparison
- Truncation
- Critical point
- Constant sign solutions
- Sign changing solutions