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5 Annales Polonici Mathematici

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Existence results for a class of quasilinear integrodifferential equations of Volterra-Hammerstein type with nonlinear boundary conditions

100%

Yang Z.

nr 3

211-218

The existence of a solution for a class of quasilinear integrodifferential equations of Volterra-Hammerstein type with nonlinear boundary conditions is established. Such equations occur in the study of the p-Laplace equation, generalized reaction-diffusion theory, non-Newtonian fluid theory, and in the study of turbulent flows of a gas in a porous medium. The results are obtained by using upper and lower solutions, and extend some previously known results.

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

63%

Yin H. , Yang Z.

2012

tom 104

nr 3

293-308

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 $Ω ⊂ ℝ^{N}$ 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 $-∆_{p(x)}u = -div(|∇u|^{p(x)-2}∇u)$ 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.

Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

63%

Yin H. , Yang Z.

nr 1

51-71

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-Δ_pu + |u|^{p-2}u = f_{1λ₁}(x) |u|^{q-2}u + 2α/(α+β) g_μ|u|^{α-2} u|v|^β$, x ∈ Ω, ⎨ $-Δ_pv + |v|^{p-2}v = f_{2λ₂}(x) |v|^{q-2}v + 2β/(α+β) g_μ|u|^α|v|^{β-2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $Ω ⊂ ℝ^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{iλ_i}(x) = λ_if_{i+}(x) + f_{i-}(x)$ (i = 1,2) are sign-changing functions, where $f_{i±}(x) = max{±f_i(x),0}$, $g_μ(x) = a(x) + μb(x)$, and $Δ_p u = div(|∇u|^{p-2}∇u)$ denotes the p-Laplace operator. We use variational methods.

Existence and nonexistence of solutions for a quasilinear elliptic system

63%

Li Q. , Yang Z.

2015

tom 113

nr 2

155-164

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-Δ_{p}u = a₁(x)F₁(x,u,v)$ in Ω, ⎪$-Δ_{q}v = a₂(x)F₂(x,u,v)$ 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 $Ω = ℝ^{N}$. A nonexistence result is obtained for radially symmetric solutions.

Asymptotics for quasilinear elliptic non-positone problems

63%

Yang Z. , Lu Q.

2002

tom 79

nr 1

85-95

In the recent years, many results have been established on positive solutions for boundary value problems of the form $-div(|∇u(x)|^{p-2} ∇u(x)) = λf(u(x))$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).