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### A nonlocal elliptic equation in a bounded domain

Banach Center Publications

The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $-{\sum }_{i,j=1}^{n}{D}_{i}\left({a}_{ij}{D}_{j}u\right)=f\left(u,{\int }_{\Omega }g\left({u}^{p}\right)\right)$, in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

### A Note on the Uniqueness and Structure of Solutions to the Dirichlet Problem for Some Elliptic Systems

In this note, we consider some elliptic systems on a smooth domain of ${R}^{n}$. By using the maximum principle, we can get a more general and complete results of the identical property of positive solution pair, and thus classify the structure of all positive solutions depending on the nonlinarities easily.

### A population biological model with a singular nonlinearity

Applications of Mathematics

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $\left\{\begin{array}{c}-{\mathrm{div}\left(|x|}^{-\alpha p}{|\nabla u|}^{p-2}{\nabla u\right)=|x|}^{-\left(\alpha +1\right)p+\beta }\left(a{u}^{p-1}-f\left(u\right)-\frac{c}{{u}^{\gamma }}\right),\phantom{\rule{1.0em}{0ex}}x\in \Omega ,\hfill \\ u=0,\phantom{\rule{1.0em}{0ex}}x\in \partial \Omega ,\hfill \end{array}\right\$ where $\Omega$ is a bounded smooth domain of ${ℝ}^{N}$ with $0\in \Omega$, $1, $0\le \alpha <\left(N-p\right)/p$, $\gamma \in \left(0,1\right)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f:\left[0,\infty \right)\to ℝ$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions...

### A variational analysis of a gauged nonlinear Schrödinger equation

Journal of the European Mathematical Society

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $-\Delta u\left(x\right)+\left(\omega +\frac{{h}^{2}\left(|x|\right)}{{|x|}^{2}}+{\int }_{|x|}^{+\infty }\frac{h\left(s\right)}{s}{u}^{2}\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds\right)u\left(x\right)={|u\left(x\right)|}^{p-1}u\left(x\right)$, where $h\left(r\right)=\frac{1}{2}{\int }_{0}^{r}s{u}^{2}\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in \left(1,3\right)$, the functional may be bounded from below or not, depending on $\omega$. Quite surprisingly, the threshold value for $\omega$ is explicit. From...

### Asymptotic behavior of ground state solution for Hénon type systems.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Blowup analysis for a semilinear parabolic system with nonlocal boundary condition.

Boundary Value Problems [electronic only]

### Comparison theorems for Riccati inequalities arising in the theory of PDEs with $p$-Laplacian.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Entire solutions for a quasilinear problem in the presence of sublinear and super-linear terms.

Boundary Value Problems [electronic only]

### Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Annales Polonici Mathematici

Our main purpose is to establish the existence of a positive solution of the system ⎧$-{∆}_{p\left(x\right)}u=F\left(x,u,v\right)$, x ∈ Ω, ⎨$-{∆}_{q\left(x\right)}v=H\left(x,u,v\right)$, x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $\Omega \subset {ℝ}^{N}$ is a bounded domain with C² boundary, $F\left(x,u,v\right)={\lambda }^{p\left(x\right)}\left[g\left(x\right)a\left(u\right)+f\left(v\right)\right]$, $H\left(x,u,v\right)={\lambda }^{q\left(x}\right)\left[g\left(x\right)b\left(v\right)+h\left(u\right)\right]$, λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $-{∆}_{p\left(x\right)}u=-{div\left(|\nabla u|}^{p\left(x\right)-2}\nabla u\right)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

### Existence and multiplicity of positive solutions to a class of quasilinear elliptic equations in ${ℝ}^{N}$.

Boundary Value Problems [electronic only]

### Existence and nonexistence of positive solutions for singular $p$-Laplacian equation in ${ℝ}^{N}$.

Boundary Value Problems [electronic only]

### Existence and nonexistence of solutions for a quasilinear elliptic system

Annales Polonici Mathematici

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁\left(x,u,v\right)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂\left(x,u,v\right)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^{N}$ with smooth boundary or $\Omega ={ℝ}^{N}$. A nonexistence result is obtained for radially symmetric solutions.

### Existence of a nonautonomous SIR epidemic model with age structure.

Advances in Difference Equations [electronic only]

### Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Annales Polonici Mathematici

We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $-\left[a+b\left({\int }_{\Omega }{|\nabla u|²dx\right)}^{m}{\right]\Delta u=f\left(x,u\right)+|u|}^{2*-2}u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $\Omega \subset {ℝ}^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

### Existence of a positive solution for a $p$-Laplacian semipositone problem.

Boundary Value Problems [electronic only]

### Existence of entire positive solutions for a class of semilinear elliptic systems.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Existence of multiple solutions for a $p\left(x\right)$-Laplace equation.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Existence of nonnegative solutions to positone-type problems in ${ℝ}^{N}$ with indefinite weights.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Existence of positive bounded solutions for some nonlinear polyharmonic elliptic systems.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Existence of positive bounded solutions of semilinear elliptic problems.

International Journal of Differential Equations

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