Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Critical points for functionals with quasilinear singular Euler–Lagrange equations

For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ .     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

Randomly weighted sums of dependent subexponential random variables*

We consider the randomly weighted sums $ \sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1} $ , where $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ are n real-valued random variables with subexponential distributions, and $ \left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\} $ are other n random variables independent of $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ and satisfying $ a \leqslant \theta \leqslant b $ for some $ 0 < a \leqslant b < \infty $ and all $ 1 \leqslant k \leqslant n $ . For $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ satisfying some dependent structures, we prove that $$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$ as x??????.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

On averages of Fourier coefficients of Maass cusp forms

Let φ be a primitive Maass cusp form and t _φ (n) be its nth Fourier coefficient at the cusp infinity. In this short note, we are interested in the estimation of the sums ${\sum_{n \leq x}t_{\varphi}(n)}$ and ${\sum_{n \leq x}t_{\varphi}(n^2)}$ . We are able to improve the previous results by showing that for any ${\varepsilon > 0}$ $$\sum_{n \leq x}t_{\varphi}(n) \ll\, _{\varphi, \varepsilon} x^{\frac{1027}{2827} + \varepsilon} \quad {and}\quad\sum_{n \leq x}t_{\varphi}(n^2) \ll\,_{\varphi, \varepsilon} x^{\frac{489}{861} + \varepsilon}.$$      

Sign-Changing Multi-Peak Solutions for Nonlinear Schrödinger Equations with Compactly Supported Potential

In this paper, we aim to prove the existence and concentration of sign-changing H ¹(? ^N ) solutions to the following nonlinear Schrödinger equations $$-\varepsilon^2\Delta u_{\varepsilon}+V(x)u_{\varepsilon}=K(x)|u_{\varepsilon}|^{p-1}u_{\varepsilon} $$ with N≥3, $1<p<\frac{N+2}{N-2}$ , and ε>0. When the potential function V(x) has compact support, $V(x)\not\equiv 0$ and V(x)≥0, K(x) is permitted to be unbounded under some necessary restrictions, we will show that one sign-changing H ¹(? ^N ) solution exists and exhibits concentration profile around local minimum points of the ground energy function $G(\xi)\equiv V^{\theta}(\xi) K^{-\frac{2}{p-1}}(\xi)$ with $\theta=\frac{p+1}{p-1}-\frac{N}{2}$ in the semiclassical limit ε→0.     

{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.     

Existence of Nodal Solutions for Dirac Equations with Singular Nonlinearities

We prove, by a shooting method, the existence of infinitely many solutions of the form ${\psi (x^{0}, x) = {{e}}^{-i \Omega x^{0}} \chi(x)}$ of the nonlinear Dirac equation $$i \underset{\mu = 0}{\overset{3}{\sum}}\gamma^{\mu} \partial_\mu \psi-m \psi - F( \overline{\psi}\psi)\psi = 0$$ where Ω >  m >  0, χ is compactly supported and $$F(x) = \left \{\begin{array}{ll}{p|x|^{p-1}} & {\rm if} |x| > 0\\ 0 & {\rm if} x = 0 \end{array}\right.$$ with ${p \in (0,1)}$ under some restrictions on the parameters p and Ω. We study also the behavior of the solutions as p tends to zero to establish the link between these equations and the M.I.T. bag model ones.     

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.     

Error function inequalities

We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$      

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.     

Infinitely many solutions for the prescribed curvature problem of polyharmonic operator

We consider the following prescribed curvature problem for polyharmonic operator: $$\left\{\begin{array}{llll} D_{m} u = \tilde{K}(y)|u|^{m^*-2}u\; {\rm in}\; \mathbb{S}^N\\ u \quad\; >0\qquad\quad\quad\quad\quad{\rm on}\; \mathbb{S}^N\\ u \quad\; \in H^{m}(\mathbb{S}^N), \end{array} \right.$$ where ${m^*=\frac{2N}{N-2m}, N\geq 2m+1,m \in \mathbb{N}_{+}, \tilde{K}}$ is positive and rationally symmetric, ${\mathbb{S}^N}$ is the unit sphere with the induced Riemannian metric ${g=g_{\mathbb{S}^N},}$ and D _m is the elliptic differential operator of 2m order given by $$\begin{array}{lll}D_m={\prod\limits_{k=1}^m}{\left(-\Delta_g+\frac{1}{4}(N-2k)(N+2k-2)\right)}\end{array}$$ where Δ _g is the Laplace-Beltrami operator on ${\mathbb{S}^N}$ . We will show that problem (P) has infinitely many non-radial positive solutions, whose energy can be arbitrary large.     

Bound states of nonlinear Schrödinger equations with magnetic fields

We study bound states of the following nonlinear Schr?dinger equation in the presence of a magnetic field: $$ \left\{\begin{array}{l} \left(-i\hbar\nabla+A(x)\right)^2u+V(x)u=g(x,|u|)u \\ |u|\in H^1(\mathbb{R}^N) \end{array} \right. $$ where ${A: \mathbb{R}^N\to\mathbb{R}^N, V: \mathbb{R}^N\to\mathbb{R}}$ and ${g: \mathbb{R}^N\times\mathbb{R}\to [0,\infty)}$ . We prove that if V is bounded below with the set ${\{x\in\mathbb{R}^N: V(x) < b\}\not=\emptyset}$ having finite measure for some b?>?0, inf V???0, and g satisfies some growth conditions, then for any integer m when ${\hbar >0 }$ is sufficiently small the problem has m geometrically different solutions.     

The reverse H?lder inequality for power means

For a function ?? non-negative on the interval [0, 1], the power mean of order ??????0 is defined by the equality $ \mathcal{M}_{\alpha \varphi} (t) = {\left( {\frac{1}{t}\int_0^t {{\varphi^\alpha }(u)du} } \right)^{1/\alpha }},\,0 < t \leqslant 1 $ . We consider the class $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ of functions ?? satisfying the reverse H?lder inequality $$ {\mathcal{M}_\beta }_\varphi \leqslant B \cdot {\mathcal{M}_\alpha }_\varphi $$ at some ???<???,??·??????0,???>?1. The sharp estimates for the summability exponents of the compositions of power means are established. As a result, we determine the properties of self-improvement of the summability exponents of functions from $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ .     

Properties of finite unrefinable chains of ring topologies

Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k.     

A reciprocity formula for quadratic forms

LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x₁,...,x₄)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ^?1 modp can be defined. Letf:?⁴→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p ₁...p _k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$      

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.