American Institute of Mathematical Sciences

We consider the following Choquard equation

\begin{document}$ \label{modelv11} \begin{cases} -(a\!+\!\varepsilon \int_{\Omega}|\nabla u|^2)\Delta u\! = \!\left( \int_{\Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^\mu}dy\right)|u|^{2^{*}_{\mu}-2}u \!+\! \lambda f(x)|u|^{q-2}u \quad in \quad \Omega,\\ u\! = \!0 \qquad \qquad \qquad \qquad \qquad on \quad \partial\Omega, \end{cases} $\end{document}

where \begin{document}$ \lambda $\end{document} is a real parameter, \begin{document}$ 2^{*}_{\mu} = \frac{2N-\mu}{N-2}(0<\mu<N) $\end{document} is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under some suitable assumptions on \begin{document}$ \lambda, \; \mu $\end{document}, via the constrained minimizer method and concentration compactness principle, we prove that this system has multiple of solutions, and one of which is a positive ground state solution. Moreover, by using an abstract result due to K.-C Chang, we admit infinitely many pairs of distinct solutions. In addition, we prove the nonexistence result by Pohožaev identity when \begin{document}$ \lambda<0 $\end{document}. The main results extend and complement the earlier works in the literature.

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form

\begin{document}$ \begin{equation*} -div (|x|^{a} D u ) = f(x,u), \; u > 0,\, \mbox{ in } \Omega, \end{equation*} $\end{document}

where \begin{document}$ N \geq 3 $\end{document}, \begin{document}$ \Omega $\end{document} is an open domain in \begin{document}$ \mathbb{R}^N $\end{document} containing the origin, \begin{document}$ N-2+a > 0 $\end{document} and \begin{document}$ f $\end{document} satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided \begin{document}$ f $\end{document} exhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the equation for \begin{document}$ f $\end{document} exhibiting the same growth. A Liouville-type theorem is then established, which asserts no positive solution of the equation in \begin{document}$ \Omega = \mathbb{R}^N $\end{document} exists provided the growth of \begin{document}$ f $\end{document} is subcritical. The results are then extended to systems of the form

\begin{document}$ \begin{equation*} -div (|x|^{a} D u_1) \! = \! f_{1}(x,u_1,u_2), -div (|x|^{a} D u_2) \! = \! f_{2}(x,u_1,u_2), u_1, u_2 \!>\! 0,\, \mbox{ in } \Omega, \end{equation*} $\end{document}

but after overcoming additional obstacles not present in the single equation. Specific cases of our results recover classical ones for a renowned problem connected with finding best constants in Hardy-Sobolev and Caffarelli-Kohn-Nirenberg inequalities as well as existence results for well-known elliptic systems.

In this paper, we show explicit $ C^{2, \alpha} $ interior estimates for viscosity solutions of fully non-linear, uniformly elliptic equations, which are close to linear equations and we also compute an explicit bound for the closeness.

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

By using the theory of maximal \begin{document}$ L^{q} $\end{document}-regularity and methods of singular analysis, we show a Taylor's type expansion–with respect to the geodesic distance around an arbitrary point–for solutions of quasilinear parabolic equations on closed manifolds. The powers of the expansion are determined explicitly by the local geometry, whose reflection to the solutions is established through the local space asymptotics.

In this paper, we proved a fractional Kirchhoff version of Hopf lemma for anti-symmetry functions and applied it to prove the symmetry and monotonicity of solutions for fractional Kirchhoff equations in the whole space by method of moving planes. We also obtain radially symmetry and monotonicity of solutions for fractional Kirchhoff equations in the unit ball. As far as we know, this is the first time to apply direct method of moving planes to fractional Kirchhoff problems.

We study the Cauchy problem for the derivative higher-order nonlinear Schrödinger equation

\begin{document}$ \begin{cases} i\partial_{t}v+\dfrac{a}{2}\partial_{x}^{2}v-\dfrac{b}{4}\partial_{x} ^{4}v = \left( \overline{\partial_{x}v}\right) ^{2},\text{ }t>1,\text{ } x\in\mathbb{R},\\ v\left( 1,x\right) = v_{0}\left( x\right) ,\text{ }x\in\mathbb{R}\text{,} \end{cases} $\end{document}

where \begin{document}$ a,b>0. $\end{document} Our aim is to prove global existence and calculate the large time asymptotics of solutions. We develop the factorization techniques originated in papers [13,10,12]. Also we follow the method of papers [9,11] to transform the quadratic nonlinearity to critical cubic nonlinearities similarly to the normal forms of Shatah [18].

We establish the uniqueness of positive radial solutions of

\begin{document}$ \begin{align} \begin{cases} \Delta u +f(u) = 0, \quad x\in A \\ u(x) = 0 \qquad \qquad x\in \partial A \end{cases} \;\;\;\; (P)\end{align} $\end{document}

where \begin{document}$ A: = A_{a, b} = \{ x\in {\mathbb R}^n : a<|x|<b \} $\end{document}, \begin{document}$ 0<a<b\le\infty $\end{document}. We assume that the nonlinearity \begin{document}$ f\in C[0, \infty)\cap C^1(0, \infty) $\end{document} is such that \begin{document}$ f(0) = 0 $\end{document} and satisfies some convexity and growth conditions, and either \begin{document}$ f(s)>0 $\end{document} for all \begin{document}$ s>0 $\end{document}, or has one zero at \begin{document}$ B>0 $\end{document}, is non positive and not identically 0 in \begin{document}$ (0, B) $\end{document} and it is positive in \begin{document}$ (B, \infty) $\end{document}.

In this paper, we study an eigenvalue problem for Schrödinger-Poisson system with indefinite nonlinearity and potential well as follows:

\begin{document}$ \begin{equation*} \begin{cases} -\Delta u+\mu V(x)u+K(x)\phi u = \lambda f(x)u+g(x)|u|^{p-2}u\ \ &\mbox{in}\ \ \mathbb{R}^3, \\ -\Delta \phi = K(x)u^2\ \ &\mbox{in}\ \ \mathbb{R}^3, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ 4\leq p<6 $\end{document}, the parameters \begin{document}$ \mu, \lambda>0 $\end{document}, \begin{document}$ V\in C(\mathbb{R}^3) $\end{document} is a potential well with the bottom \begin{document}$ \overline\Omega: = \{x\in\mathbb{R}^3 : V(x) = 0\} $\end{document}, and the functions \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} are allowed to be sign-changing. By establishing an approximate estimate between the positive principal eigenvalue of \begin{document}$ -\Delta u+\mu V(x)u = \lambda f(x)u $\end{document} in \begin{document}$ \mathbb{R}^3 $\end{document} and the positive principal eigenvalue \begin{document}$ \lambda_1(f_{\Omega}) $\end{document} of \begin{document}$ -\Delta u = \lambda f_{\Omega}(x)u $\end{document} in \begin{document}$ \Omega $\end{document}, we prove that at least a positive solution exists in \begin{document}$ 0<\lambda\leq\lambda_1(f_{\Omega}) $\end{document} while at least two positive solutions exist in \begin{document}$ \lambda>\lambda_1(f_{\Omega}) $\end{document} and near \begin{document}$ \lambda_1(f_{\Omega}) $\end{document}, where \begin{document}$ f_{\Omega}: = f|_{\Omega} $\end{document}. The results are obtained via variational method and steep potential. Furthermore, we also study the concentration of solutions as \begin{document}$ \mu\to\infty $\end{document} and the decay rate of solutions at infinity.

The objective in this paper is to study a thermal frictional contact model. The deformable body consists of a viscoelastic material and the process is assumed to be dynamic. It is assumed that the material behaves in accordance with Kelvin-Voigt constitutive law and the thermal effect is added. The variational formulation of the model leads to a coupled system including a history-dependent hemivariational inequality for the displacement field and an evolution equation for the temperature field. In study of this system, we first consider a fully discrete scheme of it and then focus on deriving error estimates for numerical solutions. Under appropriate assumptions of solution regularity, an optimal order error estimate is obtained. At the end of this manuscript, we report some numerical simulation results for the contact problem so as to verify the theoretical results.

Our aim in this paper is to prove the existence of solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a Cahn–Hilliard equation for the tumor density and a Cahn–Hilliard type equation for the oxygen concentration. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering modified equations and taking logarithmic nonlinear terms in the Cahn–Hilliard equations. After that we show a local in time weak solution which is conditionally global in time.

This paper and [20] treat the existence and nonexistence of stable (resp. outside stable) weak solutions to a fractional Hardy–Hénon equation \begin{document}$ (-\Delta)^s u = |x|^\ell |u|^{p-1} u $\end{document} in \begin{document}$ \mathbb{R}^N $\end{document}, where \begin{document}$ 0 < s < 1 $\end{document}, \begin{document}$ \ell > -2s $\end{document}, \begin{document}$ p>1 $\end{document}, \begin{document}$ N \geq 1 $\end{document} and \begin{document}$ N > 2s $\end{document}. In this paper, the nonexistence part is proved for the Joseph–Lundgren subcritical case.

To understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity, we aim to study a sixth-order Boussinesq equation with logarithmic nonlinearity in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} (\begin{document}$ n\geq1 $\end{document} is an integer) with smooth boundary \begin{document}$ \partial\Omega $\end{document}, where the dispersive and the strong damping are taken into account. Based on the Faedo-Galërkin method, the logarithmic Sobolev inequality, and the potential well method, the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow-up, and to study the polynomial decay and the exponential decay of the energy of the system.

In this paper, we study the existence of spiky stationary solutions of the Schnakenberg model with heterogeneity on compact metric graphs. These solutions are constructed by using the Liapunov–Schmidt reduction method and taking the same strategy as that in [14,11]. First, we give the abstract theorem on the existence of multi-peak solutions for general compact metric graphs under several assumptions for the associated Green's function. In particular, we reveal that how locations of concentration points and amplitudes of spiky solutions are determined by the interaction of the heterogeneity with the geometry of the compact metric graph, represented by Green's function. Second, we apply our abstract theorem to the \begin{document}$ Y $\end{document}-shaped metric graph and the \begin{document}$ H $\end{document}-shaped metric graph in non-heterogeneity case. In particular, we show the precise effect of the geometry of those compact graphs to the locations of concentration points for these concrete graphs, respectively.

In this paper, we focus on the regularity of nonnegative solutions of Lane-Emden system

\begin{document}$ \begin{equation*} \begin{cases} -\Delta u = v^p\\ -\Delta v = u^q \end{cases} \mbox{ in } \mathbb{R}^n. \end{equation*} $\end{document}

By means of Kelvin transform, we turn this problem into estimating the local integrability of \begin{document}$ (\bar{u},\bar{v}) $\end{document}. Assume that \begin{document}$ (\bar{u},\bar{v}) $\end{document} possesses some initial local integrability beforehand.

\begin{document}$ (\bar{u},\bar{v})\in L_{loc}^{r_0}(\mathbb{R}^n)\times L_{loc}^{s_0}(\mathbb{R}^n) $\end{document}

for any suitable \begin{document}$ r_0 $\end{document} and \begin{document}$ s_0 $\end{document} under specified conditions. Then through a regularity lifting method by contracting operators, we prove that

\begin{document}$ (\bar{u},\bar{v})\in L_{loc}^r(\mathbb{R}^n)\times L_{loc}^s(\mathbb{R}^n) $\end{document}

for \begin{document}$ r $\end{document} and \begin{document}$ s $\end{document} sufficiently large under twice regularity lifting if needed. Furthermore, we lift the regularity of solutions to

\begin{document}$ L^\infty(\mathbb{R}^n)\times L^\infty(\mathbb{R}^n). $\end{document}

We believe that these new methods employed in this paper can be widely applied to study a variety of other problems with different spaces and linear or nonlinear problems.

In this paper, we consider the KdV-type limit for ion dynamics system. Under the Gardner-Morikawa type transforms, we derive the KdV-type equation by the scaling \begin{document}$ \varepsilon^{\frac{1}{4}}(x-t) \rightarrow X $\end{document}, \begin{document}$ \varepsilon^{\frac{3}{4}}t\rightarrow T $\end{document} for ion dynamics system in one dimension. By proving the uniform estimates for the remainders system, we show that when \begin{document}$ \varepsilon \rightarrow 0 $\end{document}, the solutions to the ion dynamics system converge globally in time to the solutions of the KdV-type equation.

In this paper, we establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals on a compact Riemann surface \begin{document}$ ( \Sigma,g) $\end{document} with smooth boundary \begin{document}$ \partial\Sigma $\end{document}. To be exact, let \begin{document}$ \lambda_1(\partial\Sigma) $\end{document} denotes the first eigenvalue of the Laplace-Beltrami operator \begin{document}$ \Delta _ { g} $\end{document} on \begin{document}$ \partial \Sigma $\end{document}. Moreover, for any \begin{document}$ 0\leq\alpha<\lambda_1(\partial\Sigma) $\end{document}, we set \begin{document}$ \mathcal { H } = \{ u \in W^{1,2} ( \Sigma, g) : \left(\int _{\Sigma} |\nabla_g u|^2 dv_g -\alpha \int _{\partial\Sigma} {u^2}ds_g \right)^{1/2}\leq 1 \ \, \mathrm{and}\, \int _{\partial\Sigma} {u}\,ds_g = 0 \} $\end{document}, where \begin{document}$ W^{1,2}(\Sigma, g) $\end{document} is the usual Sobolev space. By the method of blow-up analysis, we first prove the supremum

\begin{document}$ \begin{equation*} \sup\limits_{ u \in \mathcal { H } }\int _ { \partial\Sigma } e ^ {\pi u^ 2} ds_g \end{equation*} $\end{document}

is attained by some function \begin{document}$ u_\alpha \in \mathcal{H}\cap C^{\infty} \left(\overline{ \Sigma}\right) $\end{document}. Further, we extend the result to the case of higher order eigenvalues. The results generalize those of Li-Liu [9] and Yang [19, 20].

We consider in \begin{document}$ \mathbb R^3 $\end{document} the singularly perturbed Schrödinger-Poisson system

\begin{document}$ \begin{equation*} \begin{cases} - \varepsilon^2\Delta v+V(x)v+\phi(x)v = f(v)\\ - \Delta\phi = v^2 . \end{cases} \end{equation*} $\end{document}

Using variational techniques, we construct solutions which concentrate around the saddle points of the external potential \begin{document}$ V $\end{document}, as \begin{document}$ \varepsilon \rightarrow 0 $\end{document}.