We obtain existence and localization results of positive nontrivial solutions for a class of semilinear elliptic variational systems. The proof is based on variants of Schechter’s localized critical point theorems for Hilbert spaces not identified to their duals and on the technique of inverse-positive matrices. The Leray–Schauder boundary condition is also involved.
P. Jebelean, R. Precup, Poincare inequalities in reflexive cones, Appl. Math. Letters 24 (2011), 359-363, http://dx.doi.org/10.1016/j.aml.2010.10.024
also freely available at the publisher: http://dx.doi.org/10.1016/j.aml.2010.10.024
Applied Mathematics Letters
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 E.A. de Barros e Silva, Existence and multiplicity of solutions for semilinear elliptic systems, NoDEA Nonlinear Differential Equations Appl. 1 (1994) 339–363.
 Ph. Clément, D.G. de Figueiredo, E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992) 923–940.
 R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal. 39 (2000) 559–568.
 D.G. De Figueiredo, Nonlinear elliptic systems, An. Acad. Bras. Ciênc. 72 (4) (2000) 453–469.
 M. Ghergu, V. Radulescu, Explosive solutions of semilinear elliptic systems with gradient term, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A ˘ Mat. 97 (3) (2003) 467–475.
 P. Jebelean, R. Precup, Solvability of p, q-Laplacian systems with potential boundary conditions, Appl. Anal. 89 (2010) 221–228.
 D. Muzsi, R. Precup, Nonresonance and existence for systems of nonlinear operator equations, Appl. Anal. 87 (9) (2008) 1005–1018.
 J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
 R. Precup, The Leray–Schauder condition in critical point theory, Nonlinear Anal. 71 (2009) 3218–3228.
 R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling 49 (2009) 703–708.
 R. Precup, Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J. Math. Anal. Appl. 352 (2009)48–56.
 R. Precup, A. Viorel, Existence results for systems of nonlinear evolution equations, Int. J. Pure Appl. Math. 47 (2008) 199–206.
 F. Robert, Matrices non-negatives et normes vectorielles (Cours de D.E.A.), Université Scientifique et Médicale, Lyon, 1973.
 M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Basel, 1999.
 J. Serrin, H. Zou, The existence of positive entire solutions of elliptic Hamiltonian systems, Comm. Partial Differential Equations 23 (1998) 577–599.