American Institute of Mathematical Sciences

We consider the slightly subcritical elliptic problem with Hardy term

\begin{document}$ \left\{ \begin{aligned} - \Delta u-\mu\frac{u}{|x|^2} & = |u|^{2^{\ast}-2- \varepsilon}u &&\quad \rm{in } \Omega\subset{\mathbb{R}}^N, \\\ u & = 0&&\quad \rm{on } \partial \Omega, \end{aligned} \right. $\end{document}

where \begin{document}$ 0\in \Omega $\end{document} and \begin{document}$ \Omega $\end{document} is invariant under the subgroup \begin{document}$ SO(2)\times\{\pm E_{N-2}\}\subset O(N) $\end{document}; here \begin{document}$ E_n $\end{document} denots the \begin{document}$ n\times n $\end{document} identity matrix. If \begin{document}$ \mu = \mu_0 \varepsilon^ \alpha $\end{document} with \begin{document}$ \mu_0>0 $\end{document} fixed and \begin{document}$ \alpha>\frac{N-4}{N-2} $\end{document} the existence of nodal solutions that blow up, as \begin{document}$ \varepsilon\to0^+ $\end{document}, positively at the origin and negatively at a different point in a general bounded domain has been proved in [5]. Solutions with more than two blow-up points have not been found so far. In the present paper we obtain the existence of nodal solutions with a positive blow-up point at the origin and \begin{document}$ k = 2 $\end{document} or \begin{document}$ k = 3 $\end{document} negative blow-up points placed symmetrically in \begin{document}$ \Omega\cap({\mathbb{R}}^2\times\{0\}) $\end{document} around the origin provided a certain function \begin{document}$ f_k:{\mathbb{R}}^+\times{\mathbb{R}}^+\times I\to{\mathbb{R}} $\end{document} has stable critical points; here \begin{document}$ I = \{t>0:(t,0,\dots,0)\in \Omega\} $\end{document}. If \begin{document}$ \Omega = B(0,1)\subset{\mathbb{R}}^N $\end{document} is the unit ball centered at the origin we obtain two solutions for \begin{document}$ k = 2 $\end{document} and \begin{document}$ N\ge7 $\end{document}, or \begin{document}$ k = 3 $\end{document} and \begin{document}$ N $\end{document} large. The result is optimal in the sense that for \begin{document}$ \Omega = B(0,1) $\end{document} there cannot exist solutions with a positive blow-up point at the origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly there do exist solutions on \begin{document}$ \Omega = B(0,1) $\end{document} with a positive blow-up point at the origin and four blow-up points on the vertices of a square with alternating positive and negative signs. The results of our paper show that the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well understood.

This paper deals with the fractional Schrödinger-Poisson system

\begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta )^su+V(x)u+K(x)\phi u = |u|^{2_{s}^{*}-2}u, & \text{in}\ {\Bbb R}^3,\\ (-\Delta)^{t}\phi = K(x)u^2, & \text{in}\ {\Bbb R}^3, \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ s\in (\frac{3}{4}, 1) $\end{document}, \begin{document}$ t\in(0, 1) $\end{document}, \begin{document}$ \varepsilon $\end{document} is a positive parameter, \begin{document}$ 2_{s}^{*} = \frac{6}{3-2s} $\end{document} is the critical Sobolev exponent. \begin{document}$ K(x)\in L^{\frac{6}{2t+4s-3}}({\Bbb R}^3) $\end{document}, \begin{document}$ V(x)\in L^{\frac{3}{2s}}({\Bbb R}^3) $\end{document} and \begin{document}$ V(x) $\end{document} is assumed to be zero in some region of \begin{document}$ {\Bbb R}^3 $\end{document}, which means that the problem is of the critical frequency case. In virtue of a global compactness result in fractional Sobolev space and Lusternik-Schnirelman theory of critical points, we succeed in proving the multiplicity of bound states.

In this paper we obtain the existence of nontrivial solutions for the fractional Laplacian equations with the nonlinearity may fail to have asymptotic limits at zero and at infinity. We make use of a combination of homotopy invariance of critical groups and the topological version of linking methods.

This paper is devoted to dealing with the existence and uniqueness of positive solutions for the following coupled nonlinear Schrödinger systems with multi-parameters

\begin{document}$ \begin{equation*} \begin{cases} &- \varDelta u = \lambda u - \mu_1 u^3 + \beta_1 uv^2,\quad \rm {in}\ \Omega,\\ &- \varDelta v = \lambda v - \mu_2 v^3 + \beta_2 u^2v,\quad \rm {in}\ \Omega,\\ &u, v > 0\quad \rm {in}\ \Omega,\quad u, v = 0\quad \rm {on}\ \partial \Omega, \end{cases} \end{equation*} $\end{document}

on the range of \begin{document}$ \lambda $\end{document} and the different coupling constants \begin{document}$ \beta_1, \beta_2 $\end{document}, where \begin{document}$ \Omega \subset \mathbb{R}^N $\end{document} \begin{document}$ (N \geqslant 1) $\end{document} is a bounded smooth domain, \begin{document}$ \lambda > 0 $\end{document} and \begin{document}$ \mu_1 \leqslant \mu_2 $\end{document}. Under some conditions, we establish some interesting positive solutions structure theorems in the \begin{document}$ \beta_1 \beta_2 $\end{document}-plane, especially we obtain the new structure theorems for the cases that \begin{document}$ \mu_1 $\end{document} and \begin{document}$ \mu_2 $\end{document} have different signs or they are negative. In addition, we get interesting uniqueness results via synchronized solutions techniques.

In this paper, we are concerned with the following generalized fully nonlinear nonlocal operators:

\begin{document}$ F_{s,m}(u(x)) = c_{N,s} m^{ \frac{N}{2}+s} P.V. \int_{\mathbb{R}^{N}} \frac{G(u(x)-u(y))}{|x-y|^{ \frac{N}{2}+s}} K_{ \frac{N}{2}+s}(m|x-y|)dy+m^{2s}u(x), $\end{document}

where \begin{document}$ s\in (0,1) $\end{document} and mass \begin{document}$ m>0 $\end{document}. By establishing various maximal principle and using the direct method of moving plane, we prove the monotonicity, symmetry and uniqueness for solutions to fully nonlinear nonlocal equation in unit ball, \begin{document}$ \mathbb{R}^{N} $\end{document}, \begin{document}$ \mathbb{R}^{N}_{+} $\end{document} and a coercive epigraph domain \begin{document}$ \Omega $\end{document} in \begin{document}$ \mathbb{R}^N $\end{document} respectively.

We consider the following fractional Schrödinger-Poisson equation with combined nonlinearities

\begin{document}$ \begin{equation*} \begin{cases} (-\Delta)^su+\phi u = |u|^{s^*-2}u+\mu|u|^{p-2}u \,\,\,\rm{in}\ \mathbb{R}^3,\\ -\Delta \phi = 4\pi u^2\ \rm{in}\ \mathbb{R}^3, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ s\in(\frac{3}{4},1) $\end{document}, \begin{document}$ \mu>0 $\end{document}, \begin{document}$ p\in(3,s^*) $\end{document} and \begin{document}$ s^* = \frac{6}{3-2s} $\end{document}. By the perturbation approach we prove the existence of the ground state solution in fractional Coulomb-Sobolev space.

The Orlicz-Minkowski problem for polytopes is studied, and some existence results are established by the variational method.

We consider a class of Chern-Simons-Schrödinger system

\begin{document}$ \begin{align*} \begin{cases} -\Delta u+V(x) u+A_{0}u+\sum\limits_{j = 1}^{2} A_{j}^{2}u = g(u), \\ \partial_{1}A_{0} = A_{2}|u|^{2}, \ \ \partial_{2}A_{0} = -A_{1}|u|^{2}, \\ \partial_{1}A_{2}-\partial_{2}A_{1} = -\frac{1}{2}u^{2}, \ \ \partial_{1}A_{1}+\partial_{2}A_{2} = 0, \end{cases} \end{align*} $\end{document}

where \begin{document}$ V $\end{document} is coercive sign-changing potential and \begin{document}$ f $\end{document} satisfies some suitable conditions. Due to lack of the mountain pass geometry and the link geometry for the corresponding variational functional, we obtain the existence of nontrivial solutions via the local link theorem.

In this paper, we prove two new improved Sobolev inequalities involving weighted Morrey norms in \begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $\end{document} and \begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $\end{document}. For instance, the corresponding inequality in \begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $\end{document} states that: there exists \begin{document}$ C\! = \!C(n, s, \alpha, \eta_1, \eta_2)\!>\!0 $\end{document} such that for each \begin{document}$ (u, v) \!\in\! {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}, \begin{document}$ p\!\in\![2, 2^*_{s}(\alpha)) $\end{document} and \begin{document}$ \theta \!\in\! (\bar{\theta}, \frac{2\eta_1}{2^*_{s}(\alpha)}) $\end{document}, it holds that

\begin{document}$ \Big( \int_{ \mathbb{R}^{n} } \frac{ |u|^{\eta_1} |v|^{\eta_2} } { |y|^{\alpha} } dy \Big)^{ \frac{1}{ 2^*_{s} (\alpha) }} \nonumber \\ \!\leq\! C ||u||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}} ||v||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}+\frac{\eta_2-\eta_1}{2^*_{s} (\alpha)}} ||(uv)||^{\frac{\eta_1}{2^*_{s} (\alpha)}-\frac{\theta}{2}}_{ L^{\frac{p}{2}, \frac{p}{2}(n-2s+r)}(\mathbb{R}^{n}, |y|^{-\frac{p}{2}r}) }, ~~~~(0)$ \end{document}

where \begin{document}$ s \!\in\! (0, 1) $\end{document}, \begin{document}$ 0\!<\!\alpha\!<\!2s\!<\!n $\end{document}, \begin{document}$ \eta_1\!+\!\eta_2\! = \!2^*_{s}(\alpha)\!: = \!\frac{2(n-\alpha)}{n-2s} $\end{document}, \begin{document}$ 1\!<\!\eta_1\!\leq\!\eta_2\!<\!\eta_1\!+\!\frac{\alpha}{s} $\end{document}, \begin{document}$ \bar{\theta}\! = \!\max \Big\{ \frac{2}{2^*_{s}(\alpha)}, \frac{2\eta_1}{2^*_{s}(\alpha)} -\frac{2t(\frac{\alpha}{2s}-\frac{\alpha}{n})}{2^*_{s}(\alpha) -\frac{2\alpha}{n}}\Big\} $\end{document}, \begin{document}$ t\! = \!1\!-\!\frac{(\eta_2-\eta_1)s}{\alpha} $\end{document} and \begin{document}$ r\! = \!\frac{2\alpha}{ 2^*_{s}(\alpha) } $\end{document}. This inequality, together with its counterpart in \begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\!\times\!{D}^{1, p}( \mathbb{R}^{n}) $\end{document} extend similar Sobolev inequality in \begin{document}$ {\dot{H}}^s( \mathbb{R}^{n}) $\end{document} as well as in \begin{document}$ {D}^{1, p}( \mathbb{R}^{n}) $\end{document} obtained by G. Palatucci and A. Pisante [Calc. Var., 50 (2014)] to the product spaces \begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $\end{document} and \begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $\end{document}, respectively.

With the help of the inequality (1), we succeed in obtaining some new existence results for doubly critical elliptic systems involving fractional Laplacian and Hardy terms.

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrödinger system

1 \begin{document}$ \begin{equation} \left\{\begin{array}{ll} -iD_0\Psi-(D_1D_1+D_2D_2)\Psi+V\Psi = |\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0 = -\frac 12i\lambda[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0 = \frac 12i\lambda[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1 = -\frac12\lambda|\Psi|^2.\\ \end{array} \right. \end{equation} $\end{document}

with an external potential \begin{document}$ V(x) $\end{document}, where \begin{document}$ D_{0} = \partial_{t}+i\lambda A_{0} $\end{document} and \begin{document}$ D_{k} = \partial_{x_k}-i\lambda A_{k}, \, k = 1,2, $\end{document} for \begin{document}$ (x_1,x_2,t)\in \mathbb{R}^{2,1} $\end{document} are covariant derivatives, \begin{document}$ \lambda $\end{document} is the coupling number. Suppose that \begin{document}$ V(x) $\end{document} satisfies \begin{document}$ \lim_{|x|\to\infty}V(x) = +\infty $\end{document}, we show for \begin{document}$ 2<p<4 $\end{document} that there exists \begin{document}$ \lambda^*>0 $\end{document} such that if \begin{document}$ 0<\lambda<\lambda^* $\end{document}, problem (1) has two nontrivial static solutions \begin{document}$ (\Psi_\lambda, A_0^\lambda, A_1^\lambda,A_2^\lambda) $\end{document}. Moreover, there also exists \begin{document}$ \tilde\lambda>0 $\end{document} such that if \begin{document}$ \lambda>\tilde\lambda $\end{document}, problem (1) has no nontrivial solutions. While for \begin{document}$ p>4 $\end{document} we assume in addition that \begin{document}$ x\cdot \nabla V(x)\geq 0 $\end{document}, then problem (1) admits a mountain pass solution for all \begin{document}$ \lambda>0 $\end{document}.

In this paper we study the existence of convex bodies for the Orlicz Minkowski problem

\begin{document}$ c\varphi (h_{K})dS(K, \cdot) = d\mu\quad \mbox{on}\, {\mathbb{S}}^{n-1} $\end{document}

where \begin{document}$ \mu $\end{document} is the given Borel measure on \begin{document}$ {\mathbb{S}}^{n-1} $\end{document}, \begin{document}$ h_{K} $\end{document} is the support function of \begin{document}$ K $\end{document}, \begin{document}$ S_{K} $\end{document} is the surface area measure of \begin{document}$ K $\end{document}, and \begin{document}$ c $\end{document} is a real parameter. We prove that, under assumptions on \begin{document}$ \varphi $\end{document} at \begin{document}$ {\it infinity} $\end{document}, there exists \begin{document}$ c_{*}>0 $\end{document} such that, if \begin{document}$ c\in [c_{*}, +\infty) $\end{document} this problem always has a solution \begin{document}$ K_{c} $\end{document}.

In this paper, we study the following \begin{document}$ k $\end{document}-coupled critical system:

\begin{document}$ \begin{equation*} \left\{ \begin{aligned} (-\Delta)^s u_i +\lambda_iu_i & = \mu_i u_i^{2^*-1}+\sum\limits_{j = 1,j\ne i}^{k} \beta_{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox{in}\;\Omega,\\ u_i & = 0 \quad \hbox{in}\;\; \; \mathbb R^N\backslash\Omega, \quad i = 1,2,\cdots, k. \end{aligned} \right. \end{equation*} $\end{document}

Here \begin{document}$ (-\Delta)^s $\end{document} is the fractional Laplacian operator, \begin{document}$ 0<s<1 $\end{document}, \begin{document}$ 2^{*} = \frac{2N}{N-2s} $\end{document} is a fractional Sobolev critical exponent, \begin{document}$ N>2s $\end{document}, \begin{document}$ - \lambda_s( \Omega)< \lambda_i<0, \mu_i>0 $\end{document}, \begin{document}$ \beta_{ij} = \beta_{ji}\ne 0 $\end{document} and \begin{document}$ \Omega\subset {\mathbb R}^N $\end{document} is a smooth bounded domain, where \begin{document}$ \lambda_s( \Omega) $\end{document} is the first eigenvalue of \begin{document}$ (-\Delta)^{s} $\end{document} with the homogeneous Dirichlet boundary datum. We characterize the positive least energy solution of the \begin{document}$ k $\end{document}-coupled fractional critical system for the purely cooperative case \begin{document}$ \beta_{ij}>0 $\end{document} with \begin{document}$ N> 4s $\end{document}. We shall introduce the idea of induction to prove our results. We point out that the key idea is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the \begin{document}$ k $\end{document}-coupled system decreases as \begin{document}$ k $\end{document} grows. Moreover, we establish the existence of positive least energy solution of the limit system in \begin{document}$ \mathbb R^N $\end{document}, as well as classification results. Meanwhile, we also construct a positive solution for a more general system involving subcritical items. Besides, we investigated in the asymptotic behaviour of the positive least energy solutions of the critical system. We point out that the results of the fractional critical systems have some coincidences with those of the critical Schrödinger systems.