Singular quasilinear convective elliptic systems in ℝ N

The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

Absolute Stability of Neutral Systems with Lurie Type Nonlinearity

The paper studies absolute stability of neutral differential nonlinear systems x ˙ ( t ) = A x t + B x t − τ + D x ˙ t − τ + b f ( σ ( t ) ) , σ ( t ) = c T x ( t ) , t ⩾ 0 $$ \begin{align}\dot x(t)=Ax\left ( t \right )+Bx\left ( {t-\tau} \right ) +D\dot x\left ( {t-\tau} \right ) +bf({\sigma (t)}),\,\, \sigma (t)=c^Tx(t), \,\, t\geqslant 0 \end{align} $$ where x is an unknown vector, A, B and D are constant matrices, b and c are column constant vectors, 𝜏 > 0 is a constant delay and f is a Lurie-type nonlinear function satisfying Lipschitz condition. Absolute stability is analyzed by a general Lyapunov-Krasovskii functional with the results compared with those previously known.

Bifurcation analysis for a modified quasilinear equation with negative exponent

In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

On non-resistive limit of 1D MHD equations with no vacuum at infinity

In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

Existence of multiple nontrivial solutions of the nonlinear Schrödinger-Korteweg-de Vries type system

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

Maximum principle for higher order operators in general domains

We first prove De Giorgi type level estimates for functions in W 1,t (Ω), Ω ⊂ R N $ \Omega\subset{\mathbb R}^N $ , with t > N ≥ 2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on ℝ N

We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝ N , which is derived from a non-Newtonain fluid for a generalized second grade fluid with Riemann-Liouville fractional derivative. We show that a solution operator involving the Laplacian operator is very effective to discuss the proposed problem. In this paper, we are concerned with the global/local well-posedness of the problem, the approaches rely on the Gagliardo-Nirenberg inequalities, operator theory, standard fixed point technique and harmonic analysis methods. We also present several results on the continuation, a blow-up alternative with a blow-up rate and the integrability in Lebesgue spaces.

Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

We study the following Kirchhoff type equation: − a + b ∫ R N | ∇ u | 2 d x Δ u + u = k ( x ) | u | p − 2 u + m ( x ) | u | q − 2 u in R N , $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N=3, a , b > 0 $ a,b \gt 0 $ , 1 < q < 2 < p < min { 4 , 2 ∗ } $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2N/(N − 2), k ∈ C (ℝ N ) is bounded and m ∈ L p/(p−q)(ℝ N ). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H 1 ( R N ) ↪ L r ( R N ) ( 2 ≤ r < 2 ∗ ) $ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

Sobolev regularity solutions for a class of singular quasilinear ODEs

This paper considers an initial-boundary value problem for a class of singular quasilinear second-order ordinary differential equations with the constraint condition stemming from fluid mechanics. We prove that the existence of positive Sobolev regular solutions for this kind of singular quasilinear ODEs by means of a suitable Nash-Moser iteration scheme Meanwhile, asymptotic expansion of those positive solutions is shown.