Abstract
We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ* > 0 such that for λ < λ*, the problem has no positive solution; for λ = λ*, it has at least one positive solution; and for λ > λ*, it has at least two positive solutions.
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Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Memoirs of the American Mathematical Society, vol. 196, no. 915, vi, 70 p. American Mathematical Society, Providence (2008)
Aizicovici S., Papageorgiou N.S., Staicu V.: Existence of multiple solutions with precise sign information for superlinear Neumann problems. Ann. Mat. Pura Appl. 188(4), 679–719 (2009)
Binding P.A., Drabek P., Huang Y.X.: Existence of multiple solutions of critical quasilinear elliptic Neumann problems. Nonlinear Anal. Ser. A Theory Methods 42(4), 613–629 (2000)
Brock F., Iturriaga L., Ubilla P.: A multiplicity result for the p-Laplacian involving a parameter. Ann. Henri Poincaré 9(7), 1371–1386 (2008)
Dong W.: A priori estimates and existence of positive solutions for a quasilinear elliptic equation. J. Lond. Math. Soc. 72, 645–662 (2005)
Garcia Azorero J.P., Peral Alonso I., Manfredi J.J.: Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2(3), 385–404 (2000)
Garcia Melian J., Sabinade Lis J.: Stationary profiles of degenerate problems when a parameter is large. Differ. Integral Equ. 13, 1201–1232 (2000)
Gasinski L., Papageorgiou N.S.: Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9. Chapman & Hall/CRC, Boca Raton (2006)
Guedda M., Veron L.: Bifurcation phenomena associated to the p-Laplace operator. Trans. Am. Math. Soc. 310, 419–431 (1988)
Guo Z.: Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems. Nonlinear Anal. 18(10), 957–971 (1992)
Guo Z., Zhang Z.: W 1, p versus C 1 local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286(1), 32–50 (2003)
Gurtin M.E., MacCamy R.C.: On the diffusion of biological populations. Math. Biosci. 33(1–2), 35–49 (1977)
Hu S., Papageorgiou N.S.: Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Commun. Pure Appl. Anal. 10(4), 1055–1078 (2011)
Kamin S., Veron L.: Flat core property associated to the p-Laplace operator. Proc. Am. Math. Soc. 118, 1079–1085 (1993)
Lieberman G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)
Logan J.D.: Introduction to Nonlinear Partial Differential Equations. Wiley-Interscience, New York (1994)
Motreanu D., Motreanu V., Papageorgiou N.S.: Multiple nontrivial solutions for nonlinear eigenvalue problems. Proc. Am. Math. Soc. 135, 3649–3658 (2007)
Motreanu D., Motreanu V., Papageorgiou N.S.: Positive solutions and multiple solutions at the non-resonance, resonance and near resonance for hemivariational inequaltiies with p-Laplacian. Trans. Am. Math. Soc. 360, 2527–2545 (2008)
Motreanu D., Motreanu V., Papageorgiou N.S.: Nonlinear Neumann problems near resonance. Indiana Univ. Math. J. 58, 1257–1279 (2009)
Motreanu D., Papageorgiou N.S.: Existence and multiplicity of solutions for Neumann problems. J. Differ. Equ. 232(1), 1–35 (2007)
Rabinowitz, P.H.: Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 23, 173–186 (1973/1974)
Takeuchi S.: Positive solutions of a degenerate elliptic equation with logistic reaction. Proc. Am. Math. Soc. 129(2), 433–441 (2001)
Takeuchi S.: Multiplicity result for a degenerate elliptic equation with logistic reaction. J. Differ. Equ. 173(1), 138–144 (2001)
Takeuchi S., Yamada Y.: Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian. Nonlinear Anal. 42, 41–61 (2000)
Vazquez J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(3), 191–202 (1984)
Wu X., Chen J.: On existence and multiplicity of solutions for elliptic equations involving the p-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 15(6), 745–755 (2008)
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Cardinali, T., Papageorgiou, N.S. & Rubbioni, P. Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type. Annali di Matematica 193, 1–21 (2014). https://doi.org/10.1007/s10231-012-0263-0
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DOI: https://doi.org/10.1007/s10231-012-0263-0
Keywords
- Superdiffusive reaction
- Neumann problem
- Truncations
- Local minimizers
- Upper and lower solutions
- p-Laplacian