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2 Annales Polonici Mathematici

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

100%

Yin H. , Yang Z.

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.

Existence of three solutions for a Navier boundary value problem involving the p(x)-biharmonic operator

100%

Yin H. , Xu M.

2013

tom 109

nr 1

47-58

The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operator with Navier boundary value conditions. The proof is mainly based on a three critical points theorem due to B. Ricceri [Nonlinear Anal. 70 (2009), 3084-3089].