On lane-emden type systems
Philip Korman Junping Shi
We consider a class of singular systems of Lane-Emden type \begin{equation} \nonumber \begin{cases} \Delta u + \la u^{p_1} v^{q_1}=0, & x\in D,\\ \Delta v + \la u^{p_2} v^{q_2}=0, & x\in D,\\ u=v=0, & x\in \partial D, \end{cases} \end{equation} with $p_1\le 0, \; p_2> 0, \; q_1> 0, \; q_2\le 0$, and $D$ a smooth domain in $\R^n$. In case the system is sublinear we prove existence of a positive solution. If $D$ is a ball in $\R^n$, we prove both existence and uniqueness of positive radially symmetric solution.
keywords: semilinear elliptic system uniqueness.
Curves of equiharmonic solutions, and problems at resonance
Philip Korman
We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu_1 \varphi_1+\cdots +\mu _n \varphi_n+e(x) \; \; for \; x \in \Omega, \; \; u=0 \; \; on \; \partial \Omega, \] where $\varphi_k$ is the $k$-th eigenfunction of the Laplacian on $\Omega$ and $e(x) \perp \varphi_k$, $k=1, \ldots,n$. Write the solution in the form $u(x)= \Sigma _{i=1}^n \xi _i \varphi_i+U(x)$, with $ U \perp \varphi_k$, $k=1, \ldots,n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\xi =(\xi _1, \ldots,\xi _n)$ fixed, but allowing for $\mu =(\mu _1, \ldots,\mu _n)$ to vary. Studying the map $\xi \rightarrow \mu$ provides us with the existence and multiplicity results for the above problem. We apply our results to problems at resonance, at both the principal and higher eigenvalues. Our approach is suitable for numerical calculations, which we implement, illustrating our results.
keywords: problems at resonance. Curves of equiharmonic solutions
On uniqueness of positive solutions for a class of semilinear equations
Philip Korman
Using the technique of Adimurthy, F. Pacella and S.L. Yadava [1], we extend an uniqueness result for a class of non-autonomous semilinear equations in M.K. Kwong and Y. Li [8]. We also observe that combining the results of [1] with bifurcation theory, one can obtain a detailed picture of the global solution curve for a class of concave-convex nonlinearities.
keywords: Uniqueness of solutions the global solution curve.

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