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We prove existence results of two solutions of the problem
where 
\[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \]
$L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, 
$p\in (2,2^{*}]$ and 
$\lambda$ and 
$m$ sufficiently large. Then their asymptotical limit as 
$m\to +\infty$ is investigated showing different behaviours.
In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem
when 
\[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \]
$(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if 
$N\geq 3$) and 
$(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if 
$N=2$), where 
$I_{\alpha }$ and 
$I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem
where 
\[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \]
$\alpha ,\,\beta \in (0,\,2)$ and 
$f,\,g$ have exponential critical growth in the Trudinger–Moser sense.
We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data have been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger convex means. Sending one of the Robin parameters to
$+\infty $
, we obtain mixed Robin/Dirichlet comparison results in shells. We prove similar results on balls and prove a comparison principle on generalized cylinders with mixed Robin/Neumann boundary conditions.
$\mathbb {R}_{+}^{3}$
In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space
under the assumption that the plane 
\[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \]
$z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.
This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in 
$\mathbb {R}^{d}$ with 
$d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in 
$\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in 
$\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.
In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function 
$\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.
This paper deals with the following fractional elliptic equation with critical exponent
where 
\[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]
$\lambda$, 
$\bar {\nu }\in {{\mathfrak R}}$, 
$s\in (0,1)$, 
$2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$, 
$(-\Delta )^{s}$ is the fractional Laplace operator, 
$\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and 
$\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition 
$u=0$ in 
${{\mathfrak R}}^{N}\setminus \Omega$. Under suitable conditions on 
$\lambda$ and 
$\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as 
$|\bar \nu | \to \infty$ .
In this study, we are concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes Korteweg equations of a compressible fluid in the half space. We assume that the space-asymptotic states and the boundary data satisfy some conditions so that the time-asymptotic state of this solution is a rarefaction wave. Then we show that the rarefaction wave is non-linearly stable, as time goes to infinity, provided that the strength of the wave is weak and the initial perturbation is small. The proof is mainly based on 
$L^{2}$-energy method and some time-decay estimates in 
$L^{p}$-norm for the smoothed rarefaction wave.
We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for 
$u(r,t)$
as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series.
We derive and numerically implement various asymptotic approximations for the lowest or principal eigenvalue of the Laplacian with a periodic arrangement of localised traps of small 
\[\mathcal{O}(\varepsilon )\] spatial extent that are centred at the lattice points of an arbitrary Bravais lattice in 
\[{\mathbb{R}^2}\]. The expansion of this principal eigenvalue proceeds in powers of 
\[\nu \equiv - 1/\log (\varepsilon {d_c})\], where dc is the logarithmic capacitance of the trap set. An explicit three-term approximation for this principal eigenvalue is derived using strong localised perturbation theory, with the coefficients in this series evaluated numerically by using an explicit formula for the source-neutral periodic Green’s function and its regular part. Moreover, a transcendental equation for an improved approximation to the principal eigenvalue, which effectively sums all the logarithmic terms in powers of v, is derived in terms of the regular part of the periodic Helmholtz Green’s function. By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved ‘log-summed’ approximation to the principal eigenvalue. From a numerical computation of the PDE eigenvalue problem defined on the fundamental Wigner–Seitz (WS) cell for the lattice, it is shown that the three-term asymptotic approximation for the principal eigenvalue agrees well with the numerical result only for a rather small trap radius. In contrast, the log-summed asymptotic result provides a very close approximation to the principal eigenvalue even when the trap radius is only moderately small. For a circular trap, the first few transcendental correction terms that further improves the log-summed approximation for the principal eigenvalue are derived. Finally, it is shown numerically that, amongst all Bravais lattices with a fixed area of the primitive cell, the principal eigenvalue is maximised for a regular hexagonal arrangement of traps.
We consider the boundary Hardy–Hénon equation
where 
\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]
$B_1(0)\subset \mathbb {R}^{N}$
$(N\geq 3)$ is a ball of radial 
$1$ centred at 
$0$, 
$p>0$ and 
$\alpha \in \mathbb {R}$. We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case 
$0< p<({N+2})/({N-2})$, we establish the estimate of positive solutions. When 
$\alpha \leq -2$ and 
$p>1$, we give some conclusions with respect to nonexistence. When 
$\alpha >-2$ and 
$1< p<({N+2})/({N-2})$, we obtain the existence of positive solution for the corresponding Dirichlet problem. When 
$0< p\leq 1$ and 
$\alpha \leq -2$, we show the nonexistence of positive solutions. When 
$0< p<1$, 
$\alpha >-2$, we give some results with respect to existence and uniqueness of positive solutions.
