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A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form - i , j = 1 n D i ( a i j D j u ) = f ( u , Ω g ( u p ) ) , 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 population biological model with a singular nonlinearity

Sayyed Hashem Rasouli (2014)

Applications of Mathematics

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form - div ( | x | - α p | u | p - 2 u ) = | x | - ( α + 1 ) p + β a u p - 1 - f ( u ) - c u γ , x Ω , u = 0 , x Ω , where Ω is a bounded smooth domain of N with 0 Ω , 1 < p < N , 0 α < ( N - p ) / p , γ ( 0 , 1 ) , and a , β , c and λ are positive parameters. Here f : [ 0 , ) 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

Alessio Pomponio, David Ruiz (2015)

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: - Δ u ( x ) + ω + h 2 ( | x | ) | x | 2 + | x | + h ( s ) s u 2 ( s ) d s u ( x ) = | u ( x ) | p - 1 u ( x ) , where h ( r ) = 1 2 0 r s u 2 ( s ) d s . 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 ( 1 , 3 ) , the functional may be bounded from below or not, depending on ω . Quite surprisingly, the threshold value for ω is explicit. From...

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

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

Our main purpose is to establish the existence of a positive solution of the system ⎧ - p ( x ) u = F ( x , u , v ) , x ∈ Ω, ⎨ - q ( x ) v = H ( x , u , v ) , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where Ω N is a bounded domain with C² boundary, F ( x , u , v ) = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] , H ( x , u , v ) = λ q ( x ) [ g ( x ) b ( v ) + h ( u ) ] , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and - p ( x ) u = - d i v ( | u | p ( x ) - 2 u ) 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 nonexistence of solutions for a quasilinear elliptic system

Qin Li, Zuodong Yang (2015)

Annales Polonici Mathematici

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧ - Δ p u = a ( x ) F ( x , u , v ) in Ω, ⎪ - Δ q v = a ( x ) F ( x , u , v ) in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in N with smooth boundary or Ω = N . A nonexistence result is obtained for radially symmetric solutions.

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

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ - [ a + b ( Ω | u | ² d x ) m ] Δ u = f ( x , u ) + | u | 2 * - 2 u in Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω 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 positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ l i m | x | u ( x ) = 0 , where Ω = x N : | x | > r , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and 0 < r r K ( r ) d r < , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the fixed point...

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