On the extremal solutions of superlinear Helmholtz problems
Abstract
Abstract: In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.
Downloads
References
I. Abid, M. Dammak and I. Douchich, Stable solutions and bifurcation problem for asymptotically linear Helmholtz equations, Nonl. Funct. Anal. and Appl, 21 (2016), 15-31.
E. Berchio, F. Gazzola, D. Pierotti, Gelfand type elliptic problems under Steklov boundary problem, Ann. I. H. Poincare -AN. Vol. 27 (2010), 315-335. https://doi.org/10.1016/j.anihpc.2009.09.011
H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for ut − ∆u = g(u) revisited, Adv. Diff. Equa. 1 (1996), 73-90.
H. Brezis, J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), 443-469. https://doi.org/10.5209/rev_REMA.1997.v10.n2.17459
M. G. Crandall, P. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal. 58 (1975), 207-218. https://doi.org/10.1007/BF00280741
H. Kielhofer, Bifurcation Theory. An Introduction with Applications to Partial Differential Equations (SpringerVerlag), Berlin, 2003.
Y. Martel Uniqueness of weak extremal solutions of nonlinear elliptic problems Houston J. Math. 23 (1997), 161-168.
P. Mironescu and V. Radulescu, The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear Anal. 26 (1996), 857-875. https://doi.org/10.1016/0362-546X(94)00327-E
Copyright (c) 2021 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
