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In this paper, we study under what boundary conditions the inequality
holds true. It is known that such an estimate holds if either the tangential or normal component of ω vanishes on the boundary ∂Ω. We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions, we give an interpolation result between these two classical boundary conditions.
$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$
In this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation where Ω is a bounded smooth domain in ℝN (N ≥ 1), α ≥ 2 and θ is a parameter. Under the assumption that g(x, u) is sublinear near the origin with respect to u, we study the effect of the perturbation term h(x, u), which may break the symmetry of the associated energy functional. With the aid of critical point theory and the truncation method, we show that this system possesses multiple small negative energy solutions.
$$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$
This paper is concerned with two frequency-dependent susceptible–infected–susceptible epidemic reaction–diffusion models in heterogeneous environment, with a cross-diffusion term modelling the effect that susceptible individuals tend to move away from higher concentration of infected individuals. It is first shown that the corresponding Neumann initial-boundary value problem in an n-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly in-time bounded regardless of the strength of the cross-diffusion and the spatial dimension n. It is further shown that, even in the presence of cross-diffusion, the models still admit threshold-type dynamics in terms of the basic reproduction number 
$\mathcal {R}_0$ – i.e. the unique disease-free equilibrium is globally stable if 
$\mathcal {R}_0\lt1$, while if 
$\mathcal {R}_0\gt1$, the disease is uniformly persistent and there is an endemic equilibrium (EE), which is globally stable in some special cases with weak chemotactic sensitivity. Our results on the asymptotic profiles of EE illustrate that restricting the motility of susceptible population may eliminate the infectious disease entirely for the first model with constant total population but fails for the second model with varying total population. In particular, this implies that such cross-diffusion does not contribute to the elimination of the infectious disease modelled by the second one.
In this paper we consider the existence of least energy nodal solution for the defocusing quasilinear Schrödinger equation
where λ≥0 is a real parameter, V(x) is a non-vanishing function, a(x) can be a vanishing positive function at infinity, the nonlinearity g(u) is of subcritical growth, the exponent p≥22*, and N≥3. The proof is based on a dual argument on Nehari manifold by employing a deformation argument and an 
$$-\Delta u - u \Delta u^2 + V(x)u = a(x)[g(u) + \lambda \vert u \vert ^{p-2}u] \hbox{in} {\open R}^N,$$
$L</italic>^{\infty}({\open R}^{N})$-estimative.
We consider the global behaviour for large solutions of the Dirac–Klein–Gordon system in critical spaces in dimension 
$1+3$. In particular, we show that bounded solutions exist globally in time and scatter, provided that a controlling space–time Lebesgue norm is finite. A crucial step is to prove nonlinear estimates that exploit the dichotomy between transversality and null structure, and furthermore involve the controlling norm.
We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces 
$H^{s}(\mathbb{T})$, 
$s>-\frac{1}{3}$, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in 
$H^{s}(\mathbb{T})$, 
$s>-\frac{9}{20}$, via the short-time Fourier restriction norm method. By following the argument in Guo–Oh for the cubic NLS, this also leads to nonexistence of solutions for the (nonrenormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in 
$H^{s}(\mathbb{T})$, 
$s>-\frac{1}{3}$, by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the 
$H^{s}$-energy functional, allowing us to introduce an infinite sequence of correction terms to the 
$H^{s}$-energy functional in the spirit of the 
$I$-method. In fact, the main novelty of this paper is this reduction of the 
$H^{s}$-energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.
We are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions in W1, p (ℝN)), for quasilinear scalar field equations of the form
where Δpu: = div(|∇ u|p−2∇u), 1 < p < N, p*: = Np/(N − p) is the critical Sobolev exponent, q ∈ (p, p*), while V(·) and K(·) are non-negative continuous potentials that may decay to zero as |x| → ∞ but are free from any integrability or symmetry assumptions.
$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$
