# American Institute of Mathematical Sciences

## Journals

CPAA
Communications on Pure & Applied Analysis 2006, 5(4): 813-826 doi: 10.3934/cpaa.2006.5.813
We consider the problem of uniqueness of radial ground state solutions to

(P) $\qquad\qquad\qquad -\Delta u=K(|x|)f(u),\quad x\in \mathbb R^n.$

Here $K$ is a positive $C^1$ function defined in $\mathbb R^+$ and $f\in C[0,\infty)$ has one zero at $u_0>0$, is non positive and not identically 0 in $(0,u_0)$, and it is locally lipschitz, positive and satisfies some superlinear growth assumption in $(u_0,\infty)$.

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CPAA
Communications on Pure & Applied Analysis 2006, 5(1): 71-84 doi: 10.3934/cpaa.2006.5.71
We consider the problem of uniqueness of radial ground state solutions to

$(P)\qquad\qquad\qquad\qquad -\Delta u=K(|x|)f(u),\quad x\in \mathbb R^n.$

Here $K$ is a positive $C^1$ function defined in $\mathbb R^+$ and $f\in C[0,\infty)$ has one zero at $u_0>0$, is non positive and not identically 0 in $(0,u_0)$, and it is locally lipschitz, positive and satisfies some superlinear growth assumption in $(u_0,\infty)$.

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PROC
Conference Publications 2003, 2003(Special): 313-319 doi: 10.3934/proc.2003.2003.313
We consider problems of the form

$(\phi(u'))' = f(t, u, u'), t \in (0, 1)$;

under the three point boundary condition

$u'(0) = 0, u(n) = u(1);$

where $n \in$ (0, 1) is given. This problem is at resonance. Three-point boundary value problems at resonance have been studied in several papers, we present here some new result as well as generalizations of some results valid for particular forms of the operator -$(\phi(u'))'. keywords: DCDS Discrete & Continuous Dynamical Systems - A 2012, 32(2): 411-432 doi: 10.3934/dcds.2012.32.411 In this work we study the nonnegative solutions of the elliptic system $\Delta u=|x|^{a}v^{\delta},\qquad\Delta v=|x|^{b}u^{\mu}%$ in the superlinear case$\mu\delta>1,$which blow up near the boundary of a domain of$\mathbb{R}^{N},$or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension$N$. We also show the existence of infinitely many solutions blowing up at$0.$Furthermore, we show that there exists a global positive solution in$\mathbb{R}^{N}\backslash\left\{ 0\right\} ,$large at$0,$and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation $\Delta^{2}u=\left\vert x\right\vert ^{b}\left\vert u\right\vert ^{\mu}.$ Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, introduced in [4], combined with the nonradial upper estimates of [5]. keywords: CPAA Communications on Pure & Applied Analysis 2013, 12(4): 1547-1568 doi: 10.3934/cpaa.2013.12.1547 In this article we study quasilinear systems of two types, in a domain$\Omega$of$R^N$: with absorption terms, or mixed terms: \begin{eqnarray} (A): \mathcal{A}_{p} u=v^{\delta}, \mathcal{A}_{q}v=u^{\mu},\\ (M): \mathcal{A}_{p} u=v^{\delta}, -\mathcal{A}_{q}v=u^{\mu}, \end{eqnarray} where$\delta$,$\mu>0$and$1 < p$,$ q < N$, and$D = \delta \mu- (p-1) (q-1) > 0$; the model case is$\mathcal{A}_p = \Delta_p$,$\mathcal{A}_q = \Delta_q.$Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type: \begin{eqnarray} u(x)\leq Cd(x,\partial\Omega)^{-\frac{p(q-1)+q\delta}{D}},\qquad v(x)\leq Cd(x,\partial\Omega)^{-\frac{q(p-1)+p\mu}{D}}. \end{eqnarray} Concerning system$(M)$, we show that$v$always satisfies Harnack inequality. In the case$\Omega=B(0,1)\backslash \{0\}\$, we also study the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.
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DCDS
Discrete & Continuous Dynamical Systems - A 2007, 19(2): 299-321 doi: 10.3934/dcds.2007.19.299
Boundary value problems for systems of ordinary differential equations are studied. These systems involve asymptotically homogeneous operators. Leray-Schauder indices are calculated for these operators and the concept of pseudo-eigenvalue is defined. The existence of nontrivial solutions is studied. Conditions for bifurcation, from either zero or infinity, at the pseudo-eigenvalues are given.
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