Existence of nontrivial solutions for p-Laplacian variational inclusion systems in ℝ N
Tóm tắt
The authors study the existence of nontrivial solutions to p-Laplacian variational inclusion systems
$\left\{ \begin{gathered}
- \Delta _p u + \left| u \right|^{p - 2} u \in \partial _1 F\left( {u,v} \right), in \mathbb{R}^N , \hfill \\
- \Delta _p v + \left| v \right|^{p - 2} v \in \partial _2 F\left( {u,v} \right), in \mathbb{R}^N , \hfill \\
\end{gathered} \right.$
where N ≥ 2, 2 ≤ p ≤ N and F: ℝ2 → ℝ is a locally Lipschitz function. Under some growth conditions on F, and by Mountain Pass Theorem and the principle of symmetric criticality, the existence of such solutions is guaranteed.
Tài liệu tham khảo
Bartsch, T. and de Figueiredo, D. G., Infinitely many solutions of nonlinear elliptic systems, Prog. Nonlinear Diff. Eqs. Appl., 35, 1999, 51–67.
Bartsch, T. and Willem, M., Infinitely many non-radial solutions of a Euclidean scalar field equation, J. Funct. Anal., 117, 1993, 447–460.
Chang, K. C., Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80, 1981, 102–129.
Clarke, F. H., Nonsmooth Analysis and Optimization, Wiley, New York, 1983.
Costa, D. G., On a class of elliptic systems in R N, Electron, J. Diff. Eqs., 111, 1994, 103–122.
de Figueiredo, D. G., Semilinear elliptic systems, Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997, World Science Publ., River Edge, New Jersey, 1998, 122–152.
Krawcewicz, W. and Marzantowicz, W., Some remarks on the Lusternik-Schnirelman method for non-differentiable functionals invariant with respect to a finite group action, Rocky Mountain J. Math., 20, 1990, 1041–1049.
Kristály, A., Existence of nonzero weak solutions for a class of elliptic variational inclusions systems in R N, Nonlinear Anal., 65, 2006, 1578–1594.
Kristály, A., Exinstemce of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains, Proc. Edinburgh Math. Soc., 48, 2005, 1–13.
Kristály, A., Lisei, H. and Varga, C., Multiple solutions for p-Laplacian type equations, Nonlinear Anal., 68, 2008, 1375–1381.
Kristály, A., Varga, C. and Varga, V., An eigenvalue problem for hemivariational inequalities with combined nonlinearities on an infinite strip, Nonlinear Anal., 63, 2005, 260–272.
Kristály, A., Varga, C. and Varga, V., A nonsmooth principle of symmetric criticality and variational-hemivariational inequalities, J. Math. Anal. Appl., 325, 2007, 975–986.
Motreanu, D. and Panagiotopoulos, P. D., Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer Academic Publishers, Dordrecht, Boston, London, 1999.
Velin, J., Existence results for some nonlinear elliptic system with lack of compactness, Nonlinear Anal., 52, 2002, 1017–1037.
Willem, M., Minimax Theorems, Birkhäuser, Boston, 1996.
Yang, M. B. and Shen, Z. F., Multiplicity result for quasilinear elliptic systems with Neumann boundary condition, Acta Anal. Funct. Appl., 7(2), 2005, 146–150.