Nonlinear Neumann problems with indefinite potential and concave terms
Shouchuan Hu Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2015, 14(6): 2561-2616 doi: 10.3934/cpaa.2015.14.2561
In this paper we conduct a detailed study of Neumann problems driven by a nonhomogeneous differential operator plus an indefinite potential and with concave contribution in the reaction. We deal with both superlinear and sublinear (possibly resonant) problems and we produce constant sign and nodal solutions. We also examine semilinear equations resonant at higher parts of the spectrum and equations with a negative concavity.
keywords: bifurcation local minimizer. nonlinear maximum principle positive solutions; nodal solutions Harnack inequality Nonlinear regularity
Flow-invariant sets and critical point theory
Jingxian Sun Shouchuan Hu
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 483-496 doi: 10.3934/dcds.2003.9.483
In this paper, we study the relationship between flow-invariant sets for an vector field $-f'(x)$ in a Banach space, and the critical points of the functional $f(x)$. The Mountain-Pass Lemma, for functionals defined on a Banach space, is generalized to a more general setting where the domain of the functional $f$ can be any flow-invariant set for $-f'(x)$. Furthermore, the intuitive approach taken in the proofs provides a new technique in proving multiple critical points.
keywords: connected set. Invariant set of flow critical point forward saturated solution
Cover page and Preface
Shouchuan Hu Xin Lu
Conference Publications 2015, 2015(special): i-i doi: 10.3934/
The Tenth AIMS International Conference on Dynamical Systems, Di eren- tial Equations and Applications took place in the magni cent Madrid, Spain, July 7 - 11, 2014. The present volume is the Proceedings, consisting of some carefully selected submissions after a rigorous refereeing process.

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Nonlinear Dirichlet problems with a crossing reaction
Shouchuan Hu Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2014, 13(6): 2749-2766 doi: 10.3934/cpaa.2014.13.2749
We consider a nonlinear Dirichlet problem driven by the sum of a $p$-Laplacian ($p>2$) and a Laplacian, with a reaction that is ($p-1$)-sublinear and exhibits an asymmetric behavior near $\infty$ and $-\infty$, crossing $\hat{\lambda}_1>0$, the principal eigenvalue of $(-\Delta_p, W^{1,p}_0(\Omega))$ (crossing nonlinearity). Resonance with respect to $\hat{\lambda}_1(p)>0$ can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction $C^1$ in the $x$-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.
keywords: Nonlinear regularity critical groups. resonance nonlinear maximum principle
Nonlinear Neumann equations driven by a nonhomogeneous differential operator
Shouchuan Hu Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2011, 10(4): 1055-1078 doi: 10.3934/cpaa.2011.10.1055
We consider a nonlinear Neumann problem driven by a nonhomogeneous nonlinear differential operator and with a reaction which is $(p-1)$-superlinear without necessarily satisfying the Ambrosetti-Rabinowitz condition. A particular case of our differential operator is the $p$-Laplacian. By combining variational methods based on critical point theory with truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive and the other negative).
keywords: nonlinear regularity Morse relation Moser iteration method. Mountain Pass theorem critical group C-condition
Positive solutions for resonant (p, q)-equations with concave terms
Shouchuan Hu Nikolas S. Papageorgiou
Communications on Pure & Applied Analysis 2018, 17(6): 2639-2656 doi: 10.3934/cpaa.2018125

We consider a parametric (p, q)-equation with competing nonlinearities in the reaction. There is a parametric concave term and a resonant Caratheordory perturbation. The resonance is with respect to the principal eigenvalue and occurs from the right. So the energy functional of the problem is indefinite. Using variational tools and truncation and comparison techniques we show that for all small values of the parameter the problem has at least two positive smooth solutions.

keywords: (p, q)-equation concave term resonant term nonlinear regularity theory strong comparison principle and variational methods
Convex solutions of boundary value problem arising from Monge-Ampère equations
Shouchuan Hu Haiyan Wang
Discrete & Continuous Dynamical Systems - A 2006, 16(3): 705-720 doi: 10.3934/dcds.2006.16.705
In this paper we study an eigenvalue boundary value problem which arises when seeking radial convex solutions of the Monge-Ampère equations. We shall establish several criteria for the existence, multiplicity and nonexistence of strictly convex solutions for the boundary value problem with or without an eigenvalue parameter.
keywords: strictly convex solution fixed index theorem. boundary value problem Monge-Ampère equation
Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems
Shouchuan Hu Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2013, 12(6): 2889-2922 doi: 10.3934/cpaa.2013.12.2889
We consider nonlinear Neumann and Dirichlet problems driven by a nonhomogeneous differential operator and a Caratheodory reaction. Our framework incorporates $p$-Laplacian equations and equations with the $(p,q)$-differential operator and with the generalized $p$-mean curvature operator. Using variational methods, together with truncation and comparison techniques and Morse theory, we prove multiplicity theorems, producing three, five or six nontrivial smooth solutions, all with sign information.
keywords: maximum principle superlinear reaction. Strong comparison principle constant sign solutions Morse theory nodal solutions
Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities
Shouchuan Hu Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2012, 11(5): 2005-2021 doi: 10.3934/cpaa.2012.11.2005
We study a semilinear parametric Dirichlet equation with an indefinite and unbounded potential. The reaction is the sum of a sublinear (concave) term and of an asymptotically linear resonant term. The resonance is with respect to any nonprincipal nonnegative eigenvalue of the differential operator. Using variational methods based on the critical point theory and Morse theory (critical groups), we show that when the parameter $\lambda>0$ is small, the problem has at least three nontrivial smooth solutions.
keywords: Indefinite and unbounded potential Morse index unique continuation property mountain pass theorem. critical groups double resonance
Positive solutions for Robin problems with general potential and logistic reaction
Shouchuan Hu Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2016, 15(6): 2489-2507 doi: 10.3934/cpaa.2016046
We consider a semilinear Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superdiffusive lotistic-type reaction. We prove bifurcation results describing the dependence of the set of positive solutions on the parameter of the problem. We also establish the existence of extreme positive solutions and determine their properties.
keywords: Robin problem. Indefinite and unbounded potential semilinear equation positive solution superdiffusive reaction bifurcation-type theorem

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