Wang, Yulan Winkler, Michael Xiang, Zhaoyin
Published in
Advances in Nonlinear Analysis

The Keller-Segel-Stokes system (*) n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n ∇ c ) + ρ n − μ n α , c t + u ⋅ ∇ c = Δ c − c + n , u t = Δ u + ∇ P − n ∇ Λ , ∇ ⋅ u = 0 , $$\begin{eqnarray*} \left\{ \begin{array}{lcll} n_t + u\cdot\nabla n &=& \it\Delta n - \nabla \cdot (n\nabla c) + \rho n - \mu n^\alpha, \\[1mm] c_t + u\cdot\nabla c &=& \it\Delta c-c+n, \\[1mm] ...

Liu, Jingjing Ji, Chao
Published in
Advances in Nonlinear Analysis

This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation (ϵi∇−A(x))2u+V(x)u+ϵ−2(|x|−1∗|u|2)u=f(|u|2)u+|u|4uin R3,u∈H1(R3,C), $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\ver...

Zeng, Shengda Bai, Yunru Gasiński, Leszek Winkert, Patrick
Published in
Advances in Nonlinear Analysis

In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by 𝓢 the solution set of the obstacle problem and by 𝓢n the solution sets of approximating problems, we prove the following convergence relation ∅≠...

Biagi, Stefano Calamai, Alessandro Marcelli, Cristina Papalini, Francesca
Published in
Advances in Nonlinear Analysis

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type ( Φ ( k ( t ) x ′ ( t ) ) ) ′ + f ( t , G x ( t ) ) ρ ( t , x ′ ( t ) ) = 0 , $$\big({\it \Phi}(k(t)\,x'(t))\big)' + f(t,{{\mathcal{G}}}_x(t))\,\rho(t, x'(t)) = 0,$$ on a compact interval [a, b]. These equations are qui...

Shen, Zupei Yu, Jianshe
Published in
Advances in Nonlinear Analysis

In this article, we consider a class of Kirchhoff equations with critical Hardy-Sobolev exponent and indefinite nonlinearity, which has not been studied in the literature. We prove very nicely that this equation has at least two solutions in ℝ3. And some known results in the literature are improved.

He, Wei Qin, Dongdong Wu, Qingfang
Published in
Advances in Nonlinear Analysis

In this paper, we study following Kirchhoff type equation: −a+b∫Ω|∇u|2dxΔu=f(u)+h in Ω,u=0 on ∂Ω. $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the ca...

Yang, Fengyan Ning, Zhen-Hu Chen, Liangbiao
Published in
Advances in Nonlinear Analysis

In this paper, we consider the following nonlinear Schrödinger equation: iut+Δgu+ia(x)u−|u|p−1u=0(x,t)∈M×(0,+∞),u(x,0)=u0(x)x∈M, $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p-1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{M},\end{cases} \end{array}$$(0.1) where (𝓜, g) is ...

Hu, Yanbo Li, Fengyan
Published in
Advances in Nonlinear Analysis

The transonic channel flow problem is one of the most important problems in mathematical fluid dynamics. The structure of solutions near the sonic curve is a key part of the whole transonic flow problem. This paper constructs a local classical hyperbolic solution for the 3-D axisymmetric steady compressible full Euler equations with boundary data g...

Zhao, Xin Zou, Wenming
Published in
Advances in Nonlinear Analysis

In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: −Δu+λ1u=μ1u3+βuv2+2qpyu2qp−1v2inΩ,−Δv+λ2v=μ2v3+βu2v+2yu2qpvinΩ,u,v>0inΩ,u,v=0on∂Ω, $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \...

Bathory, Michal Bulíček, Miroslav Málek, Josef
Published in
Advances in Nonlinear Analysis

We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity v, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus mode...