Topological degree for solutions of fourth order mean field equations

We consider the following fourth order mean field equation with Navier boundary condition $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\,\,{\rm in}\, \Omega,{\quad}u = \Delta u = 0\,\,{\rm on}\,\partial \Omega,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ where h is a C ^2,?? positive function, ?? is a bounded and smooth domain in ${\mathbb{R}^4}$ . We prove that for ${\rho \in (32m\sigma_3, 32(m + 1)\sigma_3)}$ the degree-counting formula for (*) is given by $$d(\rho)=\left\{\begin{array}{ll}\frac{1}{m!} (-\chi (\Omega) +1) \cdot\cdot \cdot (-\chi(\Omega)+m) & {\rm for}\, m >0 ,\\ 1 & {\rm for}\, m=0\end{array}\right.$$ where ??(??) is the Euler characteristic of ??. Similar result is also proved for the corresponding Dirichlet problem $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$      

Liouville Systems of Mean Field Equations

In this article, we discuss the recent work of Lin and Zhang on the Liouville system of mean field equations: $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} ({\frac{{h_j}e^{u_{j}}}{\int_{M}{h_{j}e^{u_{j}}}}-{\frac{1}{|M|}}})=0\,\, \quad{\rm on}\, M,$$ where M is a compact Riemann surface and |M| is the area, or $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} \frac{{h_j}e^{u_{j}}}{\int_{\Omega}{h_{j}e^{u_{j}}}}=0\,\, \quad{\rm in}\, \Omega,$$ $${u_j}=0,\,\, \quad{\rm on}\, \partial\Omega, $$ where ?? is a bounded domain in ${\mathbb{R}^2}$ . Among other things, we completely determine the set of non-critical parameters and derive a degree counting formula for these systems.     

Exact multiplicity results for a singularly perturbed Neumann problem

In this paper we study the number of the boundary single peak solutions of the problem $$\left\{\begin{array}{ll} -\varepsilon^{2} \Delta u + u = u^{p}, \quad {\rm in}\, \Omega \\ u > 0, \quad\quad\quad\quad\quad\quad {\rm in}\, \Omega \\ \frac{\partial u}{\partial {\nu}} = 0, \quad\quad\quad\quad\quad\,\,\, {\rm on}\, \partial {\Omega}\end{array}\right.$$ for ${\varepsilon}$ small and p subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.     

Infinitely many positive solutions for the nonlinear Schrödinger equations in {\mathbb{R}^N}

We consider the following nonlinear problem in ${\mathbb {R}^N}$ $$- \Delta u +V(|y|)u = u^{p},\quad u > 0 \quad {\rm in}\, \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \quad \quad \quad (0.1)$$ where V(r) is a positive function, ${1< p < {\frac{N+2}{N-2}}}$ . We show that if V(r) has the following expansion: $$V(r) = V_0+\frac a {r^m} +O \left(\frac1{r^{m+\theta}}\right),\quad {\rm as} \, r\to +\infty,$$ where a > 0, m > 1, θ > 0, and V ₀ > 0 are some constants, then (0.1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.     

Inequalities for the harmonic numbers

We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.     

Regularity criteria for the solutions to the 3D MHD equations in the multiplier space

In this paper, some improved regularity criteria for the 3D viscous MHD equations are established in multiplier spaces. It is proved that if the velocity field satisfies $$u \in L^{\frac{2}{1-r}}\left( 0,T,\overset{.}{X}_{r}(\mathbb{R}^{3}) \right) \quad {\rm with}\,r\in [0,1[,$$ or the gradient field of velocity satisfies $$\nabla u\in L^{\frac{2}{2-\gamma}}\left(0,T,\overset{.}{X}_{\gamma}(\mathbb{R}^{3}) \right) \quad {\rm with}\,\gamma \in \left[ 0,1\right],$$ then the solution remains smooth on [0, T].     

Radial and non radial solutions for Hardy–Hénon type elliptic systems

We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type: $$\left\{\begin{array}{ll} {-\Delta v=|x|^{\alpha}u^{p},\,-\Delta u=|x|^{\beta}v^{q} \,\,{\rm in}\, \Omega,}\\ {u=v=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad{\rm on}\, \partial \Omega}, \end{array}\right.$$ where ${\Omega\ni 0}$ is a bounded domain in ${\mathbb{R}^{N}}$ , N ≥ 3, p, q > 1, and α, β > ?N. We also study symmetry breaking for ground states when Ω is the unit ball in ${\mathbb{R}^{N}}$ .     

Asymptotic uniformity of the quantization error of self-similar measures

Let??? be a self-similar measure on ${\mathbb{R}^d}$ associated with a family of contractive similitudes {S ₁, . . . , S _N } and a probability vector {p ₁, . . . , p _N }. Let ${(\alpha_n)_{n=1}^\infty}$ be a sequence of n-optimal sets for??? of order r. For each n, we denote by ${\{P_a(\alpha_n) : a \in \alpha_n\}}$ a Voronoi partition of ${\mathbb{R}^d}$ with respect to ?? _n . Under the strong separation condition for {S ₁, . . . , S _N }, we show that the nth quantization error of??? of order ${r \in [1, \infty)}$ satisfies the following asymptotic uniformity property: $$\int \limits _{P_a(\alpha_n)}{\rm d}(x, a)^rd\mu(x) \asymp \frac{1}{n}V_{n,r}(\mu),\quad {\rm for\,all}\,a \in \alpha_n.$$      

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .     

Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.     

Generalized Moser–Trudinger inequality for unbounded domains and its application

We give a version of the Moser–Trudinger inequality for Orlicz–Sobolev spaces embedded into exponential and multiple exponential spaces on unbounded domains in ${\mathbb R^n, n \geq 2}$ . Applying this result and the Mountain Pass Theorem we study the existence of non-trivial weak solutions to the problem $$\begin{array}{ll}u \in W^1 L^{\Phi}(\mathbb R^n)\quad{\rm and}\\\quad -{\rm div} \left(\Phi ' (|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)+V(x)\Phi'(|u|)\frac{u}{|u|} =f(x,u)\quad{\rm in}\, \mathbb R^n,\end{array}$$ where Φ is a Young function such that the space ${W^1 L^{\Phi}(\mathbb R^n)}$ is embedded into an Orlicz space of the exponential or multiple exponential type, the nonlinearity f(x, t) has the corresponding critical growth and V(x) is a continuous potential.     

A remark on global existence of solutions of shadow systems

Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d ₁, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].     

Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$(-i\nabla - A(x))^2{u} = \mu|u|^{q-2}u + |u|^{2^*-2}u\, {\rm in}\, \Omega,\quad u=0\, {\rm on}\, \partial\, \Omega$$ where \({\Omega \subset \mathbb{R}^N(N \geq 4)}\) is a bounded domain with smooth boundary, \({A: \overline{\Omega} \rightarrow \mathbb{R}^N}\) is a continuous magnetic potential and \({2 \leq q < 2^* = \frac{2N}{N-2}}\) . Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of Ω.     

Large viscosity solutions for some fully nonlinear equations

We study existence, uniqueness and asymptotic behavior near the boundary of solutions of the problem $$\left\{\begin{array}{ll}-F(D^{2} u) + \beta (u) = f \quad {\rm in} \, \Omega, \\ u = + \infty \quad \quad \quad \quad \quad \quad \,\,\,\, {\rm on}\, \partial \Omega, \end{array} \right.\quad \quad \quad \quad \quad {\rm (P)}$$ where Ω is a bounded smooth domain in ${{\mathbb R}^N, N >1 , F}$ is a fully nonlinear elliptic operator and β is a nondecreasing continuous function. Assuming that β satisfies the Keller–Osserman condition, we obtain existence results which apply to ${f \in L^\infty_{loc}(\Omega)}$ or f having only local integrability properties where viscosity solutions are well defined, i.e. ${f \in L^N_{loc}(\Omega)}$ . Besides, we find the asymptotic behavior near the boundary of solutions of (P) for a wide class of functions ${f \in \mathcal{C}(\Omega)}$ . Based in this behavior, we also prove uniqueness.     

A Liouville theorem for conformal Gaussian curvature type equations in {{\mathbb{R}}^2}

In this paper, we obtain a Liouville type theorem for a class of elliptic equations including the conformal Gaussian curvature equation $$-\Delta u=K(x)e^{2u}\quad {\rm in}\,\, {\mathbb{R}}^2,$$ where K(x) is a H?lder continuous function in ${{\mathbb{R}}^2}$ that does not have a fixed sign near infinity. The main tool in our approach is an asymptotic formula for the solution at infinity and the method of moving planes. We also show how our Liouville theorem can be used to obtain a priori bound for solutions of the prescribing Gaussian curvature equation in S ², namely $$\Delta\, u+K(x)e^{2u}=1\, {\rm in}\, S^2,$$ where K(x) is H?lder continuous and nonnegative in S ² but vanishes on a set with nonempty interior, a case left open in previous research.     

On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C ¹-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters.     

A local mountain pass type result for a system of nonlinear Schr?dinger equations

We consider a singular perturbation problem for a system of nonlinear Schr?dinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$ where N?=?2, 3, ?? ₁, ?? ₂, ?? > 0 and V ₁(x), V ₂(x): R ^N ?? (0, ??) are positive continuous functions. We consider the case where the interaction ?? > 0 is relatively small and we define for ${P\in{\bf R}^N}$ the least energy level m(P) for non-trivial vector solutions of the rescaled ??limit?? problem: $$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$ We assume that there exists an open bounded set ${\Lambda\subset{\bf R}^N}$ satisfying $$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$ We show that (*) possesses a family of non-trivial vector positive solutions ${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$ which concentrates??after extracting a subsequence ?? _n ?? 0??to a point ${P_0\in\Lambda}$ with ${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$ . Moreover (v _1?? (x), v _2?? (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.     

{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms

In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, \Omega,\\ u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where ${0 < \alpha \leq a(x) \leq \beta, |b(x)| \leq \gamma, \gamma > 0, f \in L^2 (\Omega)}$ and Ω is a bounded subset of ${\mathbb{R}^N}$ with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ${W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}$ if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0.     

Recurrence in β-expansion over formal Laurent series

In this paper, we study the quantitative recurrence and hitting sets of β-transformation T _β on the unit disk I of formal Laurent series field $$E_\phi:= \{x\in I: \|T_\beta^nx - x\| < \|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\}$$ and $$F_\phi:=\{x\in I: \|T_\beta^nx-x_0\|<\|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\},$$ where x ₀ is any fixed point in I and ${\phi}$ is any positive function defined on ${\mathbb{N}}$ with ${\phi(n)\to\infty}$ as n → ∞. We completely determine the Hausdorff dimensions of these sets: $$\dim_{\rm H} E_{\phi}=\dim_{\rm H}F_\phi=\frac{1}{1+\liminf\limits_{n\to\infty}\frac{\phi (n)}{n}}.$$      

Optimality estimations for approximately midconvex functions

Let V be a convex subset of a normed space and let a nondecreasing function α : [0, ∞) → [0, ∞) be given. A function ${f : V \rightarrow \mathbb{R}}$ is called α-midconvex if $$f\left(\frac{x+y}{2} \right)\leq \frac{f(x)+f(y)}{2}+\alpha(\|x-y\|) \quad \,{\rm for}\, x,y\in V.$$ It is known (Tabor in Control Cybern., 38/3:656–669, 2009) that if ${f : V \rightarrow \mathbb{R}}$ is α-midconvex, locally bounded above at every point of V then $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+P_\alpha(\|x-y\|) \quad \,{\rm for}\, x, y \in V,t \in [0,1],$$ where ${P_\alpha(r):=\sum_{k=0}^\infty \frac{1}{2^k} \alpha(2{\rm dist}(2^kr, \mathbb{Z}))}$ for ${r \in \mathbb{R}}$ . We show that under some additional assumptions the above estimation cannot be improved.