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Topological Methods in Nonlinear Analysis

An indefinite concave-convex equation under a Neumann boundary condition II
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An indefinite concave-convex equation under a Neumann boundary condition II

Authors

  • Humberto Ramos Quoirin
  • Kenichiro Umezu

Keywords

Semilinear elliptic problem, concave-convex nonlinearity, positive solution, subcontinuum, a priori bound, bifurcation, topological method

Abstract

We proceed with the investigation of the problem $$ -\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \quad \mbox{in } \Omega, \qquad \frac{\partial u}{\partial \n} = 0\quad \mbox{on } \partial \Omega, \leqno{(\rom{P}_\lambda)} $$% where $\Omega$ is a bounded smooth domain in $\mathbb R^N$ ($N \geq2$), $1< q< 2< p$, $\lambda \in \mathbb R$, and $a,b \in C^\alpha(\overline{\Omega})$ with $0< \alpha< 1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of an unbounded subcontinuum of nontrivial nonnegative solutions of $(\rom{P}_\lambda)$. Our approach is based on {\it a priori} bounds, a regularisation procedure, and Whyburn's topological method.

References

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709.

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), 336–374.

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340.

M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180.

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525–598.

J. López-Gómez, M. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, J. Differential Equations 255 (2013), 503–523.

T. Ouyang, On the positive solutions of semilinear equations ∆u + λu − hup = 0 on the compact manifolds, Trans. Amer. Math. Soc. 331 (1992), 503–527.

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513.

H. Ramos Quoirin and K. Umezu, Positive steady states of an indefinite equation with a nonlinear boundary condition: existence, multiplicity and asymptotic profiles, Calc. Var. Partial Differential Equations 55 (2016), no. 4, paper no. 102.

H. Ramos Quoirin and K. Umezu, Bifurcation for a logistic elliptic equation with nonlinear boundary conditions: A limiting case, J. Math. Anal. Appl. 428 (2015), 1265–1285.

H. Ramos Quoirin and K. Umezu, On a concave-convex elliptic problem with a nonlinear boundary condition, Ann. Mat. Pura Appl. 195 (2016), 1833–1863.

H. Ramos Quoirin and K. Umezu, An indefinite concave-convex equation under a Neumann boundary condition I, preprint. arXiv:1603.04940

G.T. Whyburn, Topological Analysis, Second, revised edition, Princeton Mathematical Series, Vol. 23, Princeton University Press, Princeton, 1964.

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Published

2017-05-21

How to Cite

1.
QUOIRIN, Humberto Ramos and UMEZU, Kenichiro. An indefinite concave-convex equation under a Neumann boundary condition II. Topological Methods in Nonlinear Analysis. Online. 21 May 2017. Vol. 49, no. 2, pp. 739 - 756. [Accessed 8 July 2025].
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