On a class of critical Robin problems
-
Salvatore Leonardi
and
Nikolaos S. Papageorgiou
Abstract
We consider a nonlinear parametric Robin problem.
In the reaction, there are two terms, one critical and the other locally defined.
Using cut-off techniques, together with variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least three nontrivial smooth solutions all with sign information, which converge to zero in
Funding source: Ministero dell’Istruzione, dell’Universitá e della Ricerca
Award Identifier / Grant number: 2016-2018-2010–linea di intervento 2
Funding statement: This work has been supported by Piano della Ricerca 2016-2018-2010–linea di intervento 2: “Metodi variazionali ed equazioni differenzial”.
References
[1] M. Fuchs and L. Gongbao, Variational inequalities for energy functionals with nonstandard growth conditions, Abstr. Appl. Anal. 3 (1998), no. 1–2, 41–64. 10.1155/S1085337598000438Search in Google Scholar
[2] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Ser. Math. Anal. Appl. 9, Chapman & Hall/CRC, Boca Raton, 2006. Search in Google Scholar
[3] L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Springer, Cham, 2016. 10.1007/978-3-319-27817-9Search in Google Scholar
[4] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I: Theory, Kluwer Academic, Dordrecht, 1997, 10.1007/978-1-4615-6359-4Search in Google Scholar
[5] S. Leonardi, Morrey estimates for some classes of elliptic equations with a lower order term, Nonlinear Anal. 177 (2018), 611–627. 10.1016/j.na.2018.05.010Search in Google Scholar
[6] S. Leonardi and N. S. Papageorgiou, Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities, Positivity (2019), 10.1007/s11117-019-00681-5. 10.1007/s11117-019-00681-5Search in Google Scholar
[7] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. 10.1016/0362-546X(88)90053-3Search in Google Scholar
[8] Z. Liu and Z.-Q. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys. 56 (2005), no. 4, 609–629. 10.1007/s00033-005-3115-6Search in Google Scholar
[9] S. A. 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), no. 2, 815–829. 10.3934/cpaa.2013.12.815Search in Google Scholar
[10] D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 729–788. 10.2422/2036-2145.201012_003Search in Google Scholar
[11] N. S. Papageorgiou and V. D. Rădulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations 256 (2014), no. 7, 2449–2479. 10.1016/j.jde.2014.01.010Search in Google Scholar
[12] N. S. Papageorgiou and V. D. Rădulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math. 28 (2016), no. 3, 545–571. 10.1515/forum-2014-0094Search in Google Scholar
[13] N. S. Papageorgiou and V. D. Rădulescu, Infinitely many nodal solutions for nonlinear nonhomogeneous Robin problems, Adv. Nonlinear Stud. 16 (2016), no. 2, 287–299. 10.1515/ans-2015-5040Search in Google Scholar
[14] N. S. Papageorgiou and V. D. Rădulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016), no. 4, 737–764. 10.1515/ans-2016-0023Search in Google Scholar
[15] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst. 37 (2017), no. 5, 2589–2618. 10.3934/dcds.2017111Search in Google Scholar
[16] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Robin problems with indefinite linear part and competition phenomena, Commun. Pure Appl. Anal. 16 (2017), no. 4, 1293–1314. 10.3934/cpaa.2017063Search in Google Scholar
[17] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Positive solutions for nonlinear nonhomogeneous parametric Robin problems, Forum Math. 30 (2018), no. 3, 553–580. 10.1515/forum-2017-0124Search in Google Scholar
[18] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nonlinear Analysis—Theory and Methods, Springer Monogr. Math., Springer, Cham, 2019. 10.1007/978-3-030-03430-6Search in Google Scholar
[19] N. S. Papageorgiou, V. D. Rădulescu and D. Repovs, Nodal solutions for nonlinear nonhomogeneous Robin problems, Adv. Nonlinear Stud., to appear. 10.4171/RLM/831Search in Google Scholar
[20] P. Pucci and J. Serrin, The Maximum Principle, Progr. Nonlinear Differential Equations Appl. 73, Birkhäuser, Basel, 2007. 10.1007/978-3-7643-8145-5Search in Google Scholar
[21] Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 1, 15–33. 10.1007/PL00001436Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
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Articles in the same Issue
- Frontmatter
- On the distribution of zeros of derivatives of the Riemann ξ-function
- Representations of constant socle rank for the Kronecker algebra
- Abstract bivariant Cuntz semigroups II
- Curves on Segre threefolds
- 𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application
- On a class of critical Robin problems
- On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
- Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures
- Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
- Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
- Some remarks on products of sets in the Heisenberg group and in the affine group
- Codimension growth of central polynomials of Lie algebras
- Unramified Whittaker functions for certain Brylinski–Deligne covering groups
- Tilting classes over commutative rings
