Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

We study fully nonlinear elliptic equations such as $$F(D^2u) = u^p, \quad p>1,$$ in $\R^n$ or in exterior domains, where $F$ is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of $F$, that sharply characterizes the range of $p>1$ for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-V\'eron \cite{B} as well as Cutri and Leoni \cite{CL}, who found critical exponents for supersolutions in the whole space $\R^n$, in case $-F$ is Laplace's operator and Pucci's operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle.Comment: 16 pages, new existence results adde

Nonexistence of positive supersolutions of elliptic equations via the maximum principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of $\mathbb{R}^n$. The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the $p$-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.Comment: revised version, 32 page

Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions $\Phi$ and $\tilde \Phi$ of the PDE $$F(D^2\Phi) = 0 \quad {in} \mathbb{R}^n \setminus \{0 \}$$ for any positively homogeneous, uniformly elliptic operator $F$. Corresponding to $F$ are two unique scaling exponents $\alpha^*, \tilde\alpha^* > -1$ which describe the homogeneity of $\Phi$ and $\tilde \Phi$. We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation $F(D^2u) = 0$, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of $F(D^2u) = 0$ in $\mathbb{R}^n \setminus \{0 \}$ which are bounded on one side in a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two-player differential game.Comment: 35 pages, typos and minor mistakes correcte

Treatment of wastewater originating from aquaculture and biomass production in laboratory algae bioreactor using different carbon sources

The aim of present study was to explore the effect of different carbon sources on biomass accumulation in microalgae Nannochloropsis oculata and Tetraselmis chuii and their ability to remove N and P compounds during their cultivation in aquaculture wastewater. Microalgae cultivation was performed in laboratory bioreactor consisted from 500 mL Erlenmeyer flasks, containing 250 mL wastewater from semi closed recirculation aquaculture system. The cultures were maintained at room temperature (25-27ºC) on a fluorescent light with a light: dark photoperiod of 15 h: 9 h. The microalgae species were cultivated in wastewater with different carbon sources: glucose, lactose and saccharose. The growth of strains was checked for 96 h period. In the present study, N. oculata and T. chuii showed better growth in wastewater from aquaculture with saccharose carbon source during the experiment. The most effective reduce of nitrate and total nitrogen was proved in N. oculata cultivated in wastewater with glucose as carbon source. T. chuii cultivated in wastewater containing glucose showed 8.27% better cleaning effect in ammonium compared with N. oculata. T. chuii grew in wastewater with glucose as carbon source showed 19.5% better removal effect in phosphate compared with N. oculata strain

Treatment of wastewater originating from aquaculture and biomass production in laboratory algae bioreactor using different carbon sources

The aim of present study was to explore the effect of different carbon sources on biomass accumulation in microalgae Nannochloropsis oculata and Tetraselmis chuii and their ability to remove N and P compounds during their cultivation in aquaculture wastewater. Microalgae cultivation was performed in laboratory bioreactor consisted from 500 mL Erlenmeyer flasks, containing 250 mL wastewater from semi closed recirculation aquaculture system. The cultures were maintained at room temperature (25-27ºC) on a fluorescent light with a light: dark photoperiod of 15 h: 9 h. The microalgae species were cultivated in wastewater with different carbon sources: glucose, lactose and saccharose. The growth of strains was checked for 96 h period. In the present study, N. oculata and T. chuii showed better growth in wastewater from aquaculture with saccharose carbon source during the experiment. The most effective reduce of nitrate and total nitrogen was proved in N. oculata cultivated in wastewater with glucose as carbon source. T. chuii cultivated in wastewater containing glucose showed 8.27% better cleaning effect in ammonium compared with N. oculata. T. chuii grew in wastewater with glucose as carbon source showed 19.5% better removal effect in phosphate compared with N. oculata strain

Singular solutions of fully nonlinear elliptic equations and applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure

Existence and multiplicity for elliptic problems with quadratic growth in the gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.Comment: To appear in Comm. PD

Symbiotic Bright Solitary Wave Solutions of Coupled Nonlinear Schrodinger Equations

Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrodinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrodinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrodinger equations in two and three dimensional domains, we prove the existence of least energy solutions and study the location and configuration of symbiotic bright solitons. We use Nehari's manifold to construct least energy solutions and derive their asymptotic behaviors by some techniques of singular perturbation problems.Comment: to appear in Nonlinearit

Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case

We study the following nonlinear Schr\"{o}dinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, -\la_1(\om)0 and $\beta\neq 0$, where \lambda_1(\om) is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg problem. The special case N=4 was studied by the authors in (Arch. Ration. Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher dimensional case $N\ge 5$}. It is interesting that we can prove the existence of a positive least energy solution (u_\bb, v_\bb) {\it for any $\beta\neq 0$} (which can not hold in the special case N=4). We also study the limit behavior of (u_\bb, v_\bb) as $\beta\to -\infty$ and phase separation is expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing solutions} of the Brezis-Nirenberg problem, provided $N\ge 6$. In case \la_1=\la_2, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP