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

100%
Yin H., Yang Z.
Annales Polonici Mathematici
|
2011
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tom 102
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nr 1
51-71
EN
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.
2
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Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

100%
Yin H., Yang Z.
Annales Polonici Mathematici
|
2012
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tom 104
|
nr 3
293-308
EN
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.
3
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Existence of three solutions for a Navier boundary value problem involving the p(x)-biharmonic operator

100%
Yin H., Xu M.
Annales Polonici Mathematici
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2013
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tom 109
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nr 1
47-58
EN
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].
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