arXiv subjects
Publications and source records attributed to Philip Korman.
Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for periodic problems of first order. The results are applied to a population model with fishing, and to the existence and stability of limit cycles. We also describe in detail our numerical computations of curves of periodic solutions, and of limit cycles.
\[ \Delta u+g(u)=f(x) \s \mbox{for $x \in \Omega$}, \s u=0 \s \mbox{on $\partial \Omega$} \] decompose $f(x)=\mu _1 \p _1+e(x)$, where $\p _1$ is the principal eigenfunction of the Laplacian with zero boundary conditions, and $e(x) \perp \p _1$ in $L^2(\Omega)$, and similarly write $u(x)= \xi _1 \p _i+U (x)$, with $ U \perp \p _1$ in $L^2(\Omega)$. We study properties of the solution curve $(u(x),\mu _1)(\xi _1)$, and in particular its section $\mu _1=\mu _1(\xi _1)$, which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption $g'(u)<\la _2$. We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results.
We derive a generalized Pohozhaev's identity for radial solutions of $p$-Laplace equations, by using the approach in [5], thus extending the work of H. Br\'{e}zis and L. Nirenberg [2], where this identity was implicitly used for the Laplace equation.
We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.
For a class of oscillatory resonant problems, involving Dirichlet problems for semilinear PDE's on balls and rectangles in $R^n$, we show the existence of infinitely many solutions, and study the global solution set. The first harmonic of the right hand side is not required to be zero, or small. We also derive asymptotic formulas in terms of the first harmonic of solutions, and illustrate their accuracy by numerical computations. The numerical method is explained in detail.
We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function $g(u)$ is periodic of mean zero, $x \in R^n$, $r=|x|$, $\la _1$ is the principal eigenvalue of $\Delta$ on $B$. The problem has either infinitely many or finitely many solutions depending on the space dimension $n$. The situation turns out to be different for each of the following cases: $1 \leq n \leq 3$, $n=4$, $n=5$, $n=6$, and $n \geq 7$.
We study analytical and computational aspects for Dirichlet problem on the unit ball $B$: $|x|<1$ in $R^n$, modeled on the equation \[ \Delta u +\lambda \left(u^p+u^q \right)=0, \;\; \mbox{in $B$}, \;\; u=0 \s \mbox{on $\partial B$}, \] with a positive parameter $\lambda$, and $1 \frac{n}{n+2}$. This was already observed by I. Flores [6], who proved the existence of infinitely many ground state solutions. We study properties of infinitely many solution curves of this problem that are separated by these ground state solutions. We also study singular solutions (where $u(0)=\infty$), and again the Lin-Ni equation plays a special role. \medskip Super-critical equations are very challenging computationally: solutions exist only for very large $\lambda$, and curves of positive solutions make turns at very large values of $u(0)=||u||_{L^{\infty}}$. We overcome these difficulties by developing new results on singular solutions, and by using some delicate capabilities of {\em Mathematica} software.
In this note we unify the results of A.C. Lazer and P.O. Frederickson [3], A.C. Lazer [6], A.C. Lazer and D.E. Leach [7], J.M. Alonso and R. Ortega [1], and P. Korman and Y. Li [4] on periodic oscillations and unbounded solutions of nonlinear equations with linear part at resonance and periodic forcing. We give conditions for the existence and non-existence of periodic solutions, and obtain a rather detailed description of the dynamics for nonlinear oscillations at resonance, in case periodic solutions do not exist.
This paper deals with various cases of resonance, which is a fundamental concept of science and engineering. Specifically, we study the connections between periodic and unbounded solutions for several classes of equations and systems. In particular, we extend the classical Massera's theorem, dealing with periodic systems of the type \[ x'=A(t)x+f(t) \,, \] and clarify that this theorem deals with a case of resonance. Then we provide instability results for the corresponding semilinear systems, with the linear part at resonance. We also use the solution curves developed in [8], [9] to establish the instability results for pendulum-like equations, and for first-order periodic equations.
For the $p$-Laplace Dirichlet problem (where $φ(t)=t|t|^{p-2}$, $p>1$) \[ φ(u'(x))'+ f(u(x))=0 \;\;\;\; \mbox{for $-1 (p-1)\frac{f(u)}{u}>0$ for $u>γ>0$, while $\int_u^γf(t) \, dt < 0$ for all $u \in (0,γ)$. Then any positive solution, with $\max_{(-1,1)} u(x)=u(0)>γ$, is non-singular, no matter how many times $f(u)$ changes sign on $(0,γ)$. Uniqueness of solution follows.
For a class of equations generalizing the model case \[ Δ_p u-a(r)u^{p-1}+b(r)u^q=0 \; \; \mbox{in $B$}, \; \; u=0 \; \; \mbox{on $\partial B$}, \] where $B$ is the unit ball in $R^n$, $n \geq 1$, $r=|x|$, $p,q>1$, and $Δ_p$ denotes the $p$-Laplace operator, we give conditions for the existence and uniqueness of positive solution. In case $n=1$, we give a more general result.
In the classical Lotka-Volterra population models, the interacting species affect each other's growth rate. We propose an alternative model, in which the species affect each other through the limitation coefficients, rather then through the growth rates. This appears to be more realistic: the presence of foxes is not likely to diminish the fertility of rabbits, but will contribute to limiting rabbit's population. Both the cases of predation and of competition are considered, as well as competition in case of periodic coefficients. Our model becomes linear when one switches to the reciprocals of the variables. In another direction we use a similar idea to derive a multiplicity result for a class of periodic equations.
This paper provides both the theoretical results and numerical calculations of global solution curves, by continuation in global parameters. Each point on the solution curves is computed directly as the global parameter is varied, so that all of the turns that the solution curves make, as well as its different branches, appear automatically on the computer screen. For radial $p$-Laplace equations we present a simplified derivation of the regularizing transformation from P. Korman [15], and use this transformation for more accurate numerical computations. While for $p>2$ the solutions are not of class $C^2$, we show that they are of the form $w(r^{\frac{p}{2(p-1)}})$, where $w(z)$ is of class $C^2$. Bifurcation diagrams are also calculated for non-autonomous problems, and for the fourth order equations modeling elastic beams. We show that the first harmonic of the solution can also serve as a global parameter.
We show that Pinney's equation [2] with a constant coefficient can be reduced to its linear part by a simple change of variables. Also, Pinney's original solution is simplified slightly.
We obtain bounded for all $t$ solutions of ordinary differential equations as limits of the solutions of the corresponding Dirichlet problems on $(-L,L)$, with $L \rightarrow \infty$. We derive a priori estimates for the Dirichlet problems, allowing passage to the limit, via a diagonal sequence. This approach carries over to the PDE case.
We provide a surprising answer to a question raised in S. Ahmad and A.C. Lazer [2], and extend the results of that paper.
For the perturbed Gelfand's equation on the unit ball in two dimensions, Y. Du and Y. Lou [4] proved that the curve of positive solutions is exactly $S$-shaped, for sufficiently small values of the secondary parameter. We present a simplified proof and some extensions. This problem is prominent in combustion theory, see e.g., the book of J. Bebernes and D. Eberly [1].
We extend the classical Pohozaev's identity to semilinear elliptic systems of Hamiltonian type, providing a simpler approach, and a generalization, of the results of E. Mitidieri [6], R.C.A.M. Van der Vorst [14], and Y. Bozhkov and E. Mitidieri [1].