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Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves
Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential
| 1. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece |
| 2. | Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia, Department of Mathematics, University of Craiova, Street A.I. Cuza No 13, 200585 Craiova, Romania |
| 3. | Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva ploščad 16, SI-1000 Ljubljana, Slovenia |
We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $ λ<\widehat{λ}_{1}$ ($ \widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $ λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $ \bar{u}_{λ}$ and we investigate the properties of the map $ λ\mapsto\bar{u}_{λ}$.
References:
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S. Aizicovici, N. S. Papageorgiou and V. Staicu,
Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Memoirs Amer. Math. Soc., 196 (2008), vi+70 pp.
doi: 10.1090/memo/0915. |
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A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.
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H. Brezis and L. Nirenberg,
$ H^{1}$ versus $ C^{1}$ local minimizers, C.R. Acad. Sci. Paris, Sér. Ⅰ, 317 (1993), 465-472.
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G. D'Agui, S. Marano and N. S. Papageorgiou,
Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction, J. Math. Anal. Appl., 433 (2016), 1821-1845.
doi: 10.1016/j.jmaa.2015.08.065. |
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M. Filippakis and N. S. Papageorgiou,
Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1992.
doi: 10.1016/j.jde.2008.07.004. |
| [6] |
L. Gasinski and N. S. Papageorgiou,
Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
| [7] |
S. Hu and N. S. Papageorgiou,
Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
doi: 10.1007/978-1-4615-4665-8_17. |
| [8] |
S. Marano and N. S. Papageorgiou,
Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.
doi: 10.3934/cpaa.2013.12.815. |
| [9] |
D. Motreanu, V. Motreanu and N. S. Papageorgiou,
Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5. |
| [10] |
D. Mugnai,
Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3,379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 299-301.
doi: 10.1007/s00030-011-0129-y. |
| [11] |
N. S. Papageorgiou and V. D. Rǎdulescu,
Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
| [12] |
N. S. Papageorgiou and V. D. Rǎdulescu,
Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 367 (2015), 8723-8756.
doi: 10.1090/S0002-9947-2014-06518-5. |
| [13] |
N. S. Papageorgiou and V. D. Rǎdulescu,
Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Rev. Mat. Complut., 29 (2016), 91-126.
doi: 10.1007/s13163-015-0181-y. |
| [14] |
X. Wang,
Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.
doi: 10.1016/0022-0396(91)90014-Z. |
show all references
References:
| [1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu,
Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Memoirs Amer. Math. Soc., 196 (2008), vi+70 pp.
doi: 10.1090/memo/0915. |
| [2] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.
|
| [3] |
H. Brezis and L. Nirenberg,
$ H^{1}$ versus $ C^{1}$ local minimizers, C.R. Acad. Sci. Paris, Sér. Ⅰ, 317 (1993), 465-472.
|
| [4] |
G. D'Agui, S. Marano and N. S. Papageorgiou,
Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction, J. Math. Anal. Appl., 433 (2016), 1821-1845.
doi: 10.1016/j.jmaa.2015.08.065. |
| [5] |
M. Filippakis and N. S. Papageorgiou,
Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1992.
doi: 10.1016/j.jde.2008.07.004. |
| [6] |
L. Gasinski and N. S. Papageorgiou,
Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
| [7] |
S. Hu and N. S. Papageorgiou,
Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
doi: 10.1007/978-1-4615-4665-8_17. |
| [8] |
S. Marano and N. S. Papageorgiou,
Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.
doi: 10.3934/cpaa.2013.12.815. |
| [9] |
D. Motreanu, V. Motreanu and N. S. Papageorgiou,
Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5. |
| [10] |
D. Mugnai,
Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3,379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 299-301.
doi: 10.1007/s00030-011-0129-y. |
| [11] |
N. S. Papageorgiou and V. D. Rǎdulescu,
Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
| [12] |
N. S. Papageorgiou and V. D. Rǎdulescu,
Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 367 (2015), 8723-8756.
doi: 10.1090/S0002-9947-2014-06518-5. |
| [13] |
N. S. Papageorgiou and V. D. Rǎdulescu,
Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Rev. Mat. Complut., 29 (2016), 91-126.
doi: 10.1007/s13163-015-0181-y. |
| [14] |
X. Wang,
Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.
doi: 10.1016/0022-0396(91)90014-Z. |
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