# American Institute of Mathematical Sciences

## Journals

CPAA
Communications on Pure & Applied Analysis 2008, 7(4): 765-786 doi: 10.3934/cpaa.2008.7.765
We consider the following problem

$-\Delta u=\frac{\lambda}{(1-u)^2}$ in $\Omega$, $u=0$ on $\partial \Omega$, $0 < u < 1$ in $\Omega$

where $\Omega$ is a rather symmetric domain in $\mathbb R^2$. We prove that there exists a $\lambda_\star>0$ such that for $\lambda \in (0, \lambda_\star)$ the minimal solution is unique. Then we analyze the asymptotic behavior of touch-down solutions, i.e., solutions with max$_\Omega u_i (0) \to 1$. We show that after a rescaling, the solution will be asymptotically symmetric. As a consequence, we show that the branch of positive solutions must undergo infinitely many bifurcations as the maximums of the solutions on the branch go to 1 (possibly only changes of direction). This gives a positive answer to some open problems in [12]. Our result is new even in the radially symmetric case. Central to our analysis is the monotonicity formula, one-dimensional Sobloev inequality, and classification of solutions to a supercritical problem

$\Delta U=\frac{1}{U^2}\quad$ in $\mathbb R^2, U(0)=1, U(z) \geq 1.$

keywords:
DCDS-B
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1509-1523 doi: 10.3934/dcdsb.2017072
 $-Δ u=λ e^u$
in expanding annuli of ${\mathbb{R}^N}$ with $3 ≤q N ≤q 9$ is studied. To obtain the main results, we use a blow-up argument via Morse indices of the regular entire solutions of (0.1).
keywords:
CPAA
Communications on Pure & Applied Analysis 2008, 7(5): 1091-1107 doi: 10.3934/cpaa.2008.7.1091
We consider the following problem

$\Delta u=\lambda [\frac{1}{u^p}-\frac{1}{u^q}]$ in $B$, $u=\kappa \in (0,(\frac{p-1}{q-1})^{-1/(p-q)} ]$ on $\partial B$, $0 < u < \kappa$

in $B$, where $p > q > 1$ and $B$ is the unit ball in $\mathbb R^N$ ($N \geq 2$). We show that there exists $\lambda_\star>0$ such that for $0<\lambda <\lambda_\star$, the maximal solution is the only positive radial solution. Furthermore, if $2 \leq N < 2+\frac{4}{p+1} (p+\sqrt{p^2+p})$, the branch of positive radial solutions must undergo infinitely many turning points as the maxima of the radial solutions on the branch go to 0. The key ingredient is the use of a monotonicity formula.

keywords:
CPAA
Communications on Pure & Applied Analysis 2014, 13(6): 2493-2508 doi: 10.3934/cpaa.2014.13.2493
In this paper, we study the structure of solutions of a fourth order elliptic equation with a singular nonlinearity. For different boundary values $\kappa$, we establish the global bifurcation branches of solutions to the equation. More precisely, we show that $\kappa=1$ is a critical boundary value to change the structure of solutions to this problem.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2006, 14(1): 1-29 doi: 10.3934/dcds.2006.14.1
We study the degenerate logistic model described by the equation $u_t -$Δ$u=au-b(x)u^p$ with standard boundary conditions, where $p>1$, $b$ vanishes on a nontrivial subset $\Omega_0$ of the underlying bounded domain $\Omega\subset R^N$ and $b$ is positive on $\Omega_+=\Omega\setminus \overline{\Omega}_0$. We consider the difficult case where $\partial\Omega_0\cap \partial \Omega$≠$\emptyset$ and $\partial\Omega_+\cap \partial \Omega$≠$\emptyset$, and examine the asymptotic behaviour of the solutions. By a detailed study of a singularly mixed boundary blow-up problem, we obtain some basic results on the dynamics of the model.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2014, 34(6): 2561-2580 doi: 10.3934/dcds.2014.34.2561
We first obtain Liouville type results for stable entire solutions of the biharmonic equation $-\Delta^2 u=u^{-p}$ in $\mathbb{R}^N$ for $p>1$ and $3 \leq N \leq 12$. Then we consider the Navier boundary value problem for the corresponding equation and improve the known results on the regularity of the extremal solution for $3 \leq N \leq 12$. As a consequence, in the case of $p=2$, we show that the extremal solution $u^{*}$ is regular when $N =7$. This improves earlier results of Guo-Wei [21] ($N \leq 4$), Cowan-Esposito-Ghoussoub [2] ($N=5$), Cowan-Ghoussoub [4] ($N=6$).
keywords:
CPAA
Communications on Pure & Applied Analysis 2011, 10(2): 507-525 doi: 10.3934/cpaa.2011.10.507
We study the boundary value problem

$\Delta u=\lambda |x|^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1)

where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function satisfying $\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$, there is a critical power $p_c (\alpha)>0$, which is decreasing in $\alpha$, such that the branch of positive solutions possesses infinitely many bifurcation points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on the shape of the domain $\Omega$. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that $p>p_c (\alpha)$ and the Morse index of any radial solution (regular or singular) in this branch is finite provided that $0 < p \leq p_c (\alpha)$. This implies that the structure of the radial solution branch of (1) changes for $0 < p \leq p_c (\alpha)$ and $p > p_c (\alpha)$.

keywords:
CPAA
Communications on Pure & Applied Analysis 2016, 15(2): 399-412 doi: 10.3934/cpaa.2016.15.399
Structure of solutions of boundary value problems for a semilinear elliptic equation with singular nonlinearity is studied. It is seen that the structure of solutions relies on the boundary values. The global branches of solutions of the boundary value problems are established. Moreover, some Liouville type results for the entire solutions of the equation are also obtained.
keywords:
DCDS-B
Discrete & Continuous Dynamical Systems - B 2018, 23(10): 4187-4205 doi: 10.3934/dcdsb.2018132

By a new type of comparison principle for a fourth order elliptic problem in general domains, we investigate the structure of positive solutions to Navier boundary value problems of a perturbed fourth order elliptic equation with negative exponent, which arises in the study of the deflection of charged plates in electrostatic actuators in the modeling of electrostatic micro-electromechanical systems (MEMS). It is seen that the structure of solutions relies on the boundary values. The global branches of solutions to the Navier boundary value problems are established. We also show that the behaviors of these branches are relatively "stable" with respect to the Navier boundary values.

keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2007, 19(2): 271-298 doi: 10.3934/dcds.2007.19.271
We show that for small $\epsilon>0$, the boundary blow-up problem

$-\epsilon^2\Delta u= u (u-a(x))(1-u) \mbox{ in } \Omega, u|_{\partial\Omega}=\infty$

has solutions with sharp interior layers and spikes, apart from boundary layers. We also determine the location of these layers and spikes.

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