# Philip Korman

## Professor

### French Hall

5504

A&S Mathematical Sciences - 0025

## Professional Summary

https://math.uc.edu/~kormanp/

## Education

Ph.D., New York University 1981

## Research Support

Investigators:Philip Korman 06-01-2012 -05-2015 NSF Global Solution Curves for Systems of Elliptic Boundary Value Problems Role:PI Pending Type:Grant Level:National

## Abbreviated Publications

### Peer Reviewed Publications

A global solution curve for a class of free boundary value problems arising in plasma physics. Appl. Math. Optim. 71 (2015), no. 1, 25-38.

A global solution curve for a class of periodic problems, including the relativistic pendulum. *Appl. Anal.* 93 (2014), no. 1, 124–136.

(with Li, Yi) Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities. *J. Differential Equations* 257 (2014), no. 10,

Global solution curves for self-similar equations. *J. Differential Equations* 257 (2014), no. 7, 2543–2564.

Curves of equiharmonic solutions, and problems at resonance. *Discrete Contin. Dyn. Syst.* 34 (2014), no. 7, 2847–2860.

(with Li, Yi) Harmonic oscillators at resonance, perturbed by a non-linear friction force. *Acta Math. Sci. Ser. B Engl. Ed.* 34 (2014), no. 4,

Exact multiplicity and numerical computation of solutions for two classes of non-autonomous problems with concave-convex nonlinearities.Nonlinear Analysis 93 (2013), 226-235.

(With Yi Li) "Harmonic oscillators at resonance, perturbed by a non-linear friction force", *Acta Math. Sci. Ser. B Engl. Ed.* 34 (2014), no. 4, 1025–1028.

Nonlinear perturbations of linear elliptic systems at resonance.*Proc. Amer. Math. Soc.* 140 (2012), no. 7, 2447–2451.

(with Y. Li) A computer assisted study of uniqueness of ground state solutions. *J. Comput. Appl. Math.* 236 (2012), no. 11, 2838–2843.

*Direction of bifurcation for some non-autonomous problems*, Electronic J. of Differential Equations (EJDE) Vol. 2012 (2012), No. 214, pp. 1-13.

Adamowicz, Tomasz; Korman, Philip Remarks on time map for quasilinear equations. *J. Math. Anal. Appl.* 376 (2011), no. 2, 686–695.

Existence and Uniqueness of Solutions for a Class of

p-Laplace Equations on a Ball. Advanced Nonlinear Studies 11 (2011), 875–888.

Curves of equiharmonic solutions and solvability of elliptic systems. Differential and Integral Equations.** V olume 24, Numbers 11-12, November/December 2011.**

Curves of equiharmonic solutions, and ranges of nonlinear equations. *Adv. Differential Equations* 14 (2009), no. 9-10, 963–984.

Global solution curves for boundary value problems, with linear part at resonance. *Nonlinear Anal.* 71 (2009), no. 7-8, 2456–2467.

Korman, Philip Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems. *J. Differential Equations* 244 (2008), no. 10, 2602–2613.

An oscillatory bifurcation from infinity, and from zero. *NoDEA Nonlinear Differential Equations Appl.* 15 (2008), no. 3, 335–345.

A global approach to ground state solutions. *Electron. J. Differential Equations* ** 2008, **No. 122, 13 pp.

Korman, Philip; Li, Yi; Ouyang, Tiancheng Verification of bifurcation diagrams for polynomial-like equations. *J. Comput. Appl. Math.* 212 (2008), no. 2, 187–193.

An exact multiplicity result for a class of symmetric problems. *Port. Math.* 65 (2008), no. 2, 257–260

Korman, Philip Stability and instability of solutions of semilinear problems. *Appl. Anal.* 86 (2007), no. 2, 135–147.

Korman, Philip Computation of radial solutions of semilinear equations. *Electron. J. Qual. Theory Differ. Equ.* ** 2007, **No. 13, 14 pp. (electronic).

Korman, Philip A global solution curve for a class of periodic problems, including the pendulum equation. *Z. Angew. Math. Phys.* 58 (2007), no. 5, 749–766.

Hou, Xiaojie; Korman, Philip; Li, Yi On the direction of pitchfork bifurcation. * Proceedings of the Sixth Mississippi State–UBA Conference on Differential Equations and Computational Simulations, * 153–158, Electron. J. Differ. Equ. Conf., 15, *Southwest Texas State Univ., San Marcos, TX,* 2007.

Galstyan, A., & Li, Y. (2006). On the oscillations of the solution curve for a class of semilinear equations. *J. Math. Anal. Applic.*, *321*, 576-588.

(2006). Uniqueness and exact multiplicity of solutions of non-autonomous Dirichlet problems. *Advanced Nonlinear Studies*, *6*, 461-481.

Ouyang, T. (2006). Positivity for the linearized problem for semilinear equations. *Topol. Methods Nonlinear Anal.*, *28*(1), 53-60.

Li, Yi, & Ouyang, T. (2005). Computing the location and the direction of bifurcation. *Mathematical Research Letters*, *12*(5 & 6), 933-944.

(2005). On a principle of predatory exclusion. *Applicable Analysis*, *84*(7), 707-712.

Shi, J. (2005). On Lane-Emden type systems. *Discrete Contin. Dyn. Syst.* 510-517.

(2004). Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. *Proc. Roy. Soc. Edinburgh Sect. A*, *134*(1), 179-190.

(2003). An accurate computation of the global solution curve for the Gelfand
problem through a two point approximation. *Appl.Math and Computations*, *139*(2 & 3), 363-369.

(2003). Curves of positive solutions for supercritical problems. *Applicable Analysis*, *82*, 45-54.

Li, Y., & Ouyang, T. (2003). Perturbation of global solution curves for semilinear problems. *Adv. Nonlinear Stud.*, *3*(2), 289-299.

(2002). On uniqueness of positive solutions for a class of semilinear problems. *Discrete Contin. Dynam. Systems*, *8*(4), 865-871.

(2002). Curves of sign-changing solutions for semilinear equations. *Nonlinear Analysis TMA*, *51*(5), 801-820.

(2002). Exact multiplicity of solutions for a class of Neumann problems. *Comm. Appl. Nonlinear Anal.*, *9*(4), 1-17.

(2002). Global solution curves for semilinear systems. *Math. Methods Appl. Sci*, *25*(1), 3-20.

(2002). A remark on the non-degeneracy condition. *Results in Mathematics*, *41*(3 & 4), 334–336.

(2002). Stability and Morse indices of solutions for two classes of semilinear problems. *Comm. Appl. Nonlinear Anal.*, *9*(4), 1-17.

(2001). Monotone approximations of unstable solutions. *J. Comput. Appl. Math.*, *136*, 309-315.

(2001). On multiplicity of solutions of semilinear equations. *Mathematische Nachrichten*, *229*, 119-127.

Shi, J. (2001). New exact multiplicity results with an application to a population model. *Proc. Royal Soc. Edinburgh Ser. A.*, *131*(5), 1167-1182.

Shi, J. (2000). Instability and exact multiplicity of solutions of semilinear equations. *Proceedings of the Conference on Nonlinear Differential Equations* 311-322.

Li, Y. (2000). Infinitely many solutions at a resonance. *Electron. J. Differ. Equ. Conf.*, *5*.

(1999). Multiplicity of positive solutions for semilinear equations on circular
domains. *Comm. Appl. Nonlinear Anal.*, *6*(3), 17-35.

Li, Y. (1999). Generalized averages for solutions of two-point Dirichlet problems. *J. Math. Anal. Appl.*, *239*(2), 478-484.

Ouyang, T. (1999). Multiplicity and Morse indices of sign-changing solutions for semilinear equations
. *Canad. Appl. Math. Quart.*, *7*(3), 239-250.

(1999). Exact multiplicity of positive solutions for a class of semilinear equations on a ball. *Electron. J. Qual. Theory Differ. Equ.*, *8*.

(1998). The global solution set for a class of semilinear problems. *J. Math. Anal. Appl.*, *226*, 101-120.

(1998). A global solution curve for a class of semilinear equations. *Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations* 119–127.

(1998). Remarks on Nagumo’s condition. *Portugaliae Mathematica*, *55*(1), 1-9.

(1998). Uniqueness and exact multiplicity results for two classes of semilinear problems. *Nonlinear Analysis TMA*, *31*, 849-865.

Li, Y., & Ouyang, T. (1997). An exact multiplicity result for a class of semilinear equations. *Commun. in PDE*, *22*, 661-684.

(1997). Solution curves for semilinear equations on a ball. *Proc. of Amer. Math. Soc.*, *125*, 1997-2005.

(1997). Steady states and long time behavior of some convective reaction-diffusion equations
. *Funkcialaj Ekvacioj*, *40*, 165-183.

Lazer, A.C., & Li, Y. (1997). On homoclinic and heteroclinic orbits for Hamiltonian systems. *Differential and Integral Equations*, *10*, 357-368.

Ouyang, T. (1996). Solution curves for two classes of boundary-value problems. *Nonlinear Analysis Apllic.*, *27*, 1031-1047.

Li, Y., & Ouyang, T. (1996). Exact multiplicity results for boundary value problems with nonlinearities generalizing cubic. *Proc. Royal Soc. Edinburgh Ser. A.*, *126A*, 599-616.

Ouyang, T. (1995). Exact multiplicity results for a class of boundary value problems with cubic nonlinearities
. *J. Math. Anal. Applic.*, *194*, 328-341.

Ouyang, T. (1995). On computation of solution curves for semilinear elliptic problems
. *Numer. Funct. Anal. and Optimiz.*, *16*(1 & 2), 219-231.

(1995). Symmetry of positive solutions for elliptic problems in one dimension. *Applicable Analysis*, *58*, 351-365.

Ouyang, T. (1995). Exact multiplicity results for two classes of periodic. *J. Math. Anal. Applic.*, *194*, 763-779.

Ouyang, T. (1995). Multiplicity results for two classes of boundary-value problems. *SIAM J. of Math. Analysis*, *26*(1), 180-189.

(1994). On the dynamics of two classes of periodic ecological models. *J. of Computational and Appl. Math.*, *52*, 267-275.

Lazer, A.C. (1994). Homoclinic orbits for a class of symmetric Hamiltonian systems. *EJDE Volumes*.

Ouyang, T. (1993). Exact multiplicity results for two classes of boundary value problems. *Diff. and Integral Equations*, *6*(6), 1507-1517.

(1993). An Algorithmfor Computing Unstable Solutions of Semilinear Boundary Value Problems. *Computing*, *51*(3 & 4), 327-334.

(1992). A Computer Assisted Study of Periodic Parabolic Problems. *Appl. Math. & Computation*, *50*(2 & 3), 203-221.

(1992). Dynamics of the Lotka-Volterra Systems with Diffusion. *Applicable Analysis*, *44*(3 & 4), 191-208.

Choudury, G. (1992). On Computation of Solutions of Fully Nonlinear Elliptic Problems. *J. of Computational and Appl. Math*, *41*, 301-311.

(1992). On Periodic Solutions of Singular Perturbation Problems. *Proceedings of the Royal Society of Edinburgh*, *120A*, 143-152.

(1992). Some New Results on the Periodic Competition Model. *J. Math. Anal. Applic.*, *171*(1), 131-138.

Barbu, V. (1991). Approximating Optimal Controls of Elliptic Obstacle Problem by Monotone Iteration Schemes. *Numer. Funct. Anal. and Optimiz.*, *12*(5 & 6), 429-442.

Choudury, G. (1991). On Computation of Solutions of Nonlinear Boundary Value Problems. *Comput. Math. Appl.*, *22*(8), 49-55.

Leung, A., & Stojanovic, S. (1990). Monotone Iterations for Nonlinear
Obstacle Problems
. *J. Austr. Math. Soc. Ser.*, *B 30*, 259-276.

(1990). On Existence of Periodic Solutions for a Class of Quasilinear Non-
Coercive Problems. *Funkcialaj Ekvacioj*, *33*, 127-138.

(1990). On Periodic Solutions for Singular Perturbation Problems. *Nonlinear Analysis TMA*, *15*(5), 467-478.

Li, Y. (1999). The exactness of S-shaped bifurcation curve. *Proc. Amer. Math. Soc.*, *127*, 1011-1020.

(1989). On Computation of Solutions of Elliptic Systems. *Numer. Funct. Anal. and Optimiz.*, *10*(9 & 10), 977-990.

(1989). On Existence of Solutions for a Class of Fully Nonlinear Noncoercive Problems. *J. Math. Anal. Applic.*, *137*, 477-484.

(1989). On Existence of Solutions for a Class of Non-Coercive Problems. *Commun. in PDE*, *14*(4), 519-539.

(1989). A MaximumPrinciple for Fourth Order Ordinary Differential Equations. *Applicable Analysis*, *33*, 267-273.

(1988). On Blow-Up of Solutions of Nonlinear Evolution Equations. *Proc. Amer. Math. Soc.*, *103*, 189-197.

(1988). Computation of Displacements for Nonlinear Elastic Beam Models Using Monotone Iterations. *Intern. J. Math. and Math. Sci.*, *11*, 121-128.

(1988). On Existence of Solutions for Several Classes of Free Boundary Problems. *SIAM J. Math. Anal.*, *19*, 814-823.

(1987). On the Existence and Uniqueness of Positive Steady States in the Volterra-Lotka Ecological Models with Diffusion
. *Applicable Analysis*, *26*, 145-160.

Leung, A. (1986). A General Monotone Scheme for Elliptic Systems with
Applications to Ecological Models
. *Proc. Royal Soc. Edinburgh*, *102A*, 315-325.

(1986). Existence and Approximation of Solutions of Inverse-Positive Problems. *Nonlinear Analysis TMA*, *10*, 863-872.

(1986). Existence of Solutions for a Class of Semilinear Noncoercive Problems. *Nonlinear Analysis TMA*, *10*, 1471-1476.

(1984). On Application of Monotone Iteration Scheme to Nonlinear Non-Coercive Problems. *Nonlinear Analysis TMA*, *8*, 97-105.

(1983). Existence of Periodic Solutions for a Class of Nonlinear Problems. *Non-linear Analysis TMA*, *7*, 873-879.

(1983). Existence of Solutions for a Class of Nonlinear Non-Coercive Problems. *Comm. Partial Diff. Eqs.*, *8*, 819-846.

Non-existence of solutions for non-autonomous elliptic systems of Hamiltonian type, *Comm. Appl. Nonlinear Anal.* 21 (2014), no. 3, 13–20.

(with Li, Yi) Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities. *J. Differential Equations* 257 (2014), no. 10,

**Regularization of radial solutions of $p$-Laplace equations, and computations using infinite series. ** *Electron. J. Qual. Theory Differ. Equ.* 2015,

Non-singular solutions of two-point problems, with multiple changes of sign in the nonlinearity. *Proc. Amer. Math. Soc.* 144 (2016), no. 6, 2539–2546.

**Nonlinear elliptic equations and systems with linear part at resonance **Electron. J. Diff. Equ. Vol. 2016 (2016), No. 67, pp. 1-17.

### Book

Lectures on Differential Equations. A Textbook. Submitted for publication to Springer

Global solution curves for semilinear elliptic equations.*World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,* 2012.

### Review

MathSciences Review:

Ma, Ruyun; Xu, Jia; Han, Xiaoling Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight. *Nonlinear Anal.* 74 (2011), no. 10, 3379–3385. (Reviewer: Philip Korman).

MathSciences Review:

Hung, Kuo-Chih; Wang, Shin-Hwa Bifurcation diagrams of a *J. Math. Anal. Appl.* 375 (2011), no. 1, 294–309. (Reviewer: Philip Korman).

### Work in Progress

A global solution curve for a class of free boundary value problems arising in plasma physics. Submitted for publication.

### Book Chapter

(2006). Handbook of Differential Equations, Ordinary Differential Equations. In A. Canada, P. Drabek and A. Fonda (Eds.), *Global solution branches and exact multiplicity of solutions for two point boundary value problems* (pp. 547-606). North Holland: Elsevier Science.

### Other Publication

(2002). The global solution set for a class of semilinear problems. *Pure and Appl. Math.*, *225*. New York: Dekker.

(2001). Similarity of solution branches for two-point semilinear problems. *Proceedings of the Fifth Mississippi State Conference on Differential Equations and Computational Simulations*. Mississippi State.

## Presentations

### Invited Presentations

(10-19-2012. )
*Resonance in nonlinear equations and systems *.Wright State University, Dayton, OH. Other Institution. .
Level:State

(03-22-2016. )
*Some Special Equations *.Joint Analysis/PDE seminar, Ohio State University. Other Institution. .
Level:National

### Lecture

UC. Level:Department

UC. Level:Department

### Symposium

(03-09-2011. ) UC. Level:International

## Service

Reviewer Type:Editorial Service Level:International BAD DATE -BAD DATE

Journal of Differential Equations (Refereed two papers ) Peer Review/Referee Type:Editorial Service Level:International 03-10-2015

Journal of Differential Equations Peer Review/Referee Type:Editorial Service Level:International 02-28-2013

Mathematische Nachrichten Peer Review/Referee Type:Editorial Service Level:International 02-04-2013

Peer Review/Referee Type:Editorial Service Level:International 01-28-2013

Peer Review/Referee Type:Editorial Service Level:International 01-12-2013

Journal of Differential Equations Peer Review/Referee Type:Editorial Service Level:International 01-07-2013

MathSciences Reviewer Type:Editorial Service Level:International 02-12-2012

Nonlinear Analysis, TMA Peer Review/Referee Type:Editorial Service Level:International 09-21-2011 -01-13-2012

Mathematica Slovaca (Slovakia, published by Springer Verlag) Editorial Board Member Type:Editorial Service Level:International 01-01-2010

Communications on Applied Nonlinear Analysis Editorial Board Member Type:Editorial Service Level:International

Electronic Journal of Differential Equations, Problem Section Editorial Board Member Type:Editorial Service Level:International

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(RPT, Colloquium Coordinator, Math Bowl, Calculus Contest, PDE preliminary exams. I wrote several letters of recommendation. ) Member Type:Departmental Service Level:Department

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Reviewed a paper in Mathematical Modelling and Analysis for MathSciences Reviewer Type:Editorial Service Level:International 05-10-2015 -05-29-2015

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## Keywords

Differential Equations

## Courses Taught

-MATH-1061 CALCULUS I Level:Undergraduate

15-MATH-553 PDE & FOURIER ANAL Level:Both

15-MATH-273 DIFFERENTIAL EQNS

15-MATH-251 CALCULUS I Level:Undergraduate

15-MATH-629 PART DIFFER EQUATNS Level:Graduate

15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate

APP LIN ALGEBRA Level:Both

ORDINARY DIFF EQ Level:Graduate

15-MATH-568 APPL COMPLX ANALYS Level:Both

15-MATH-553 PDE & FOURIER ANAL Level:Both

15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate

15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate

15-MATH-554 PDE & FOURIER ANAL Level:Both

15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate

Applied ODE Level:Both

15-MATH-252 CALCULUS II Level:Undergraduate

15-MATH-352 LINEAR ALGEBRA II Level:Undergraduate

15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate

15-MATH-264 CALC & AN GEOM IV Level:Undergraduate

15-MATH-568 APPL COMPLX ANALYS Level:Both

15-MATH-355 INTRO ORD DIFF EQU Level:Undergraduate

15-MATH-273 DIFFERENTIAL EQNS Level:Undergraduate