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Positive solutions of nonlinear Robin eigenvalue problems. (English) Zbl 1355.35082

Summary: We consider a nonlinear eigenvalue problem driven by the \(p\)-Laplacian with Robin boundary condition. Using variational methods and truncation techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter \(\lambda\) varies. We also produce extremal positive solutions and study their properties.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B09 Positive solutions to PDEs
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[1] Aizicovici, Sergiu; Papageorgiou, Nikolaos S.; Staicu, Vasile, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196, 915, vi+70 pp. (2008) · Zbl 1165.47041 · doi:10.1090/memo/0915
[2] Cardinali, Tiziana; Papageorgiou, Nikolaos S.; Rubbioni, Paola, Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type, Ann. Mat. Pura Appl. (4), 193, 1, 1-21 (2014) · Zbl 1327.35109 · doi:10.1007/s10231-012-0263-0
[3] Dong, Wei, A priori estimates and existence of positive solutions for a quasilinear elliptic equation, J. London Math. Soc. (2), 72, 3, 645-662 (2005) · Zbl 1207.35155 · doi:10.1112/S0024610705006848
[4] Gasi{\'n}ski, Leszek; Papageorgiou, Nikolaos S., Nonlinear analysis, Series in Mathematical Analysis and Applications 9, xii+971 pp. (2006), Chapman & Hall/CRC, Boca Raton, FL · Zbl 1086.47001
[5] Gasi{\'n}ski, Leszek; Papageorgiou, Nikolaos S., Bifurcation-type results for nonlinear parametric elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 142, 3, 595-623 (2012) · Zbl 1242.35035 · doi:10.1017/S0308210511000126
[6] Hu, Shouchuan; Papageorgiou, Nikolas S., Handbook of multivalued analysis. Vol. I, {\rm Theory}, Mathematics and its Applications 419, xvi+964 pp. (1997), Kluwer Academic Publishers, Dordrecht · Zbl 0887.47001 · doi:10.1007/978-1-4615-6359-4
[7] Lieberman, Gary M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12, 11, 1203-1219 (1988) · Zbl 0675.35042 · doi:10.1016/0362-546X(88)90053-3
[8] Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th., Handbook of applied analysis, Advances in Mechanics and Mathematics 19, xviii+793 pp. (2009), Springer, New York · Zbl 1189.49003 · doi:10.1007/b120946
[9] Papageorgiou, Nikolaos S.; R{\u{a}}dulescu, Vicen{\c{t}}iu D., Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256, 7, 2449-2479 (2014) · Zbl 1287.35010 · doi:10.1016/j.jde.2014.01.010
[10] Papageorgiou, Nikolaos S.; R{\u{a}}dulescu, Vicen{\c{t}}iu D., Positive solutions for nonlinear nonhomogeneous Neumann equations of superdiffusive type, J. Fixed Point Theory Appl., 15, 2, 519-535 (2014) · Zbl 1311.35091 · doi:10.1007/s11784-014-0176-1
[11] Papageorgiou, Nikolaos S.; R{\u{a}}dulescu, Vicen{\c{t}}iu D., Positive solutions for perturbations of the eigenvalue problem for the Robin \(p\)-Laplacian, Ann. Acad. Sci. Fenn. Math., 40, 1, 255-277 (2015) · Zbl 1325.35131 · doi:10.5186/aasfm.2015.4011
[12] Papageorgiou, Nikolaos S.; R{\u{a}}dulescu, Vicen{\c{t}}iu D., Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, Discrete Contin. Dyn. Syst., 35, 10, 5003-5036 (2015) · Zbl 1335.35069 · doi:10.3934/dcds.2015.35.5003
[13] Takeuchi, Shingo, Positive solutions of a degenerate elliptic equation with logistic reaction, Proc. Amer. Math. Soc., 129, 2, 433-441 (electronic) (2001) · Zbl 0964.35066 · doi:10.1090/S0002-9939-00-05723-3
[14] Takeuchi, Shingo, Multiplicity result for a degenerate elliptic equation with logistic reaction, J. Differential Equations, 173, 1, 138-144 (2001) · Zbl 1033.35041 · doi:10.1006/jdeq.2000.3914
[15] Winkert, Patrick, \(L^\infty \)-estimates for nonlinear elliptic Neumann boundary value problems, NoDEA Nonlinear Differential Equations Appl., 17, 3, 289-302 (2010) · Zbl 1193.35070 · doi:10.1007/s00030-009-0054-5
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