Increasing radial solutions for Neumann problems without growth restrictions

Bonheure, Denis; Noris, Benedetta; Weth, Tobias

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 573-588.

Nous étudions lʼexistence de solutions radiales positives croissantes de problèmes de Neumann super linéaires dans la boule. Nous nʼimposons aucune restriction de croissance sur la non linéarité à lʼinfini et nos hypothèses permettent également une interaction avec le spectre. Notre approche combinne des arguments topologiques et variationnels. Nous contournons le manque de compacité en travaillant dans le cône des fonctions radiales, positives et croissantes de $H^1\left(B\right)$.

We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by considering the cone of nonnegative, nondecreasing radial functions of $H^1\left(B\right)$.

[1] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$, Progr. Math. vol. 240, Birkhäuser Verlag, Basel (2006) | MR | Zbl

[2] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381 | MR | Zbl

[3] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 no. 4 (1976), 620-709 | MR | Zbl

[4] T.B. Benjamin, A unified theory of conjugate flows, Philos. Trans. R. Soc. Lond. Ser. A 269 (1971), 587-643 | MR | Zbl

[5] V. Barutello, S. Secchi, E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl. 341 no. 1 (2008), 720-728 | MR | Zbl

[6] D. Bonheure, C. Grumiau, C. Troestler, Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions, preprint. | MR

[7] D. Bonheure, E. Serra, Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth, NoDEA Nonlinear Differential Equations Appl. 18 (2011), 217-235 | MR | Zbl

[8] M. Del Pino, M. Musso, A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 1 (2005), 45-82 | EuDML | Numdam | MR | Zbl

[9] M. Grossi, B. Noris, Positive constrained minimizers for supercritical problems in the ball, Proc. Amer. Math. Soc. 140 (2012), 2141-2154 | MR | Zbl

[10] M.A. KrasnoselʼSkiĭ, Fixed points of cone-compressing or cone-extending operators, Soviet Math. Dokl. 1 (1960), 1285-1288 | MR | Zbl

[11] M.A. KrasnoselʼSkiĭ, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen (1964) | MR

[12] M.K. Kwong, On Krasnoselʼskiĭʼs cone fixed point theorem, Fixed Point Theory Appl. 18 (2008) | EuDML | MR

[13] R.D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations II, J. Differential Equations 14 (1973), 360-394 | MR | Zbl

[14] S.I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f\left(u\right)=0$, Dokl. Akad. Nauk SSSR 165 (1965), 36-39 | MR

[15] S. Secchi, Increasing variational solutions for a nonlinear p-Laplace equation without growth conditions, Ann. Mat. Pura Appl., doi:10.1007/s10231-011-0191-4. | MR

[16] E. Serra, P. Tilli, Monotonicity constraints and supercritical Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), 63-74 | Numdam | MR | Zbl

[17] S.J. Li, Z.Q. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math. 81 (2000), 373-396 | MR | Zbl