On a semilinear fourth order elliptic problem with asymmetric nonlinearity. (English) Zbl 1509.31015

Summary: In this work, we address the existence of solutions for a biharmonic elliptic equation with homogeneous Navier boundary condition. The problem is asymmetric and has linear behavior on \(-\infty\) and superlinear on \(+\infty \). To obtain the results we apply topological methods.


31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35J40 Boundary value problems for higher-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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[1] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007. · Zbl 1125.47052
[2] E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations 229 (2006), 1-23. · Zbl 1142.35016
[3] H. Brezis and R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), 601-614. · Zbl 0358.35032
[4] Ph. Clement, D.G. de Figueiredo and E. Mitidieri, A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities, Nonlinear Partial Differential Equations (Fes, 1994), Pitman Res. Notes Math. Ser., vol. 343, Longman, Harlow, 1996, pp. 73-91. · Zbl 0868.35012
[5] F. Cuccu and G. Porru, Optimization of the first eigenvalue in problems involving the bi-Laplacian. Differ. Equ. Appl. 1 (2009), 219-235. · Zbl 1181.35154
[6] M. Cuesta and C. De Coster, A resonant-superlinear elliptic problem revisited, Adv. Nonlinear Stud. 13 (2013), 97-114. · Zbl 1277.35177
[7] M. Cuesta and C. De Coster, Superlinear critical resonant problems with small forcing term, Calc. Var. Partial Differential Equations 54 (2015), 349-363. · Zbl 1323.35022
[8] M. Cuesta, D.G. de Figueiredo and P.N. Srikanth, On a resonant-superlinear elliptic problem, Calc. Var. Partial Differential Equations 17 (2003), 221-233. · Zbl 1257.35088
[9] D.G. de Figueiredo and J.P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations 17 (1992), 339-346. · Zbl 0777.35042
[10] F.O. de Paiva and W. Rosa, Neumann problems with resonance in the first eigenvalue, Math. Nachr. 290 (2017), 2198-2206. · Zbl 1377.35071
[11] F. Gazzola, H.C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, 1991, Springer-Verlag, Berlin, 2010. · Zbl 1239.35002
[12] R. Kannan and R. Otega, Superlinear elliptic boundary value problems, Czechoslovak Math. J. 37 (1987), 386-399. · Zbl 0668.35032
[13] J.R. Ward, Perturbations with some superlinear growth for a class of second order elliptic boundary value problems, Nonlinear Anal. 6 (1982), 367-374. · Zbl 0533.35033
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