A Spectral Gap Estimate and Applications

We consider the Schrödinger operator

$$ \text{-} \frac{d^{2}}{d x^{2}} + V {\text{on an interval}}~~[a,b]~{\text{with Dirichlet boundary conditions}},$$

where V is bounded from below and prove a lower bound on the first eigenvalue λ ₁ in terms of sublevel estimates: if w _V (y) = |{x ∈ [a, b] : V (x) ≤ y}|, then

$$\lambda_{1} \geq \frac{1}{250} \min\limits_{y > \min V}{\left( \frac{1}{w_{V}(y)^{2}} + y\right)}.$$

The result is sharp up to a universal constant if {x ∈ [a, b] : V(x) ≤ y} is an interval for the value of y solving the minimization problem. An immediate application is as follows: let \({\Omega } \subset \mathbb {R}^{2}\) be a convex domain and let \(u:{\Omega } \rightarrow \mathbb {R}\) be the first eigenfunction of the Laplacian ? Δ on Ω with Dirichlet boundary conditions on ?Ω. We prove

$$\| u \|_{L^{\infty}({\Omega})} \lesssim \frac{1}{\text{inrad}({\Omega})} \left( \frac{\text{inrad}({\Omega})}{\text{diam}({\Omega})} \right)^{1/6} \|u\|_{L^{2}({\Omega})},$$

which answers a question of van den Berg in the special case of two dimensions.

     

Geometric constant term functor(s)

We study the Eisenstein series and constant term functors in the framework of geometric theory of automorphic functions. Our main result says that for a parabolic \(P\subset G\) with Levi quotient M, the !-constant term functor

$$\begin{aligned}{\text {CT}}_!:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M)\end{aligned}$$

is canonically isomorphic to the *-constant term functor

$$\begin{aligned} {\text {CT}}^-_*:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M), \end{aligned}$$

taken with respect to the opposite parabolic \(P^-\).

     

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|~(p-2)?u) =-|u|~(β-1) u + α|u|~(q-2 )u,where p 1, β 0, q≥1 and α 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.     

An existence principle for solutions to a singular boundary-value problems

The singular boundary-value problem

$ \left\{ {\begin{array}{*{20}{c}} {{u^{\prime\prime}} + g\left( {t,u,{u^{\prime}}} \right) = 0\quad {\text{for}}\quad t \in \left( {0,1} \right),} \hfill \\ {u(0) = u(1) = 0} \hfill \\ \end{array} } \right. $

is studied. The singularity may appear at u?=?0, and the function g may change sign. An existence theorem for solutions to the above boundary-value problem is proposed, and it is proved via the method of upper and lower solutions.

     

Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if

$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$

or

$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$

where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).

     

Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$

Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent:

$$\begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned}$$

(0.1)

For \(N\ge 4\) and \(\sigma \in (\frac{1}{2}, 1),\) we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).

     

Impulsive fractional boundary value problem with <Emphasis Type="Italic">p</Emphasis>-Laplace operator

In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem

$$\begin{aligned} {_{t}}D_{T}^{\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) + a(t)|u(t)|^{p-2}u(t)= & {} f(t,u(t)),\;\;t\ne t_j,\;\;\hbox {a.e.}\;\;t\in [0,T],\\ \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} I_j(u(t_j))\;\;j=1,2,\ldots ,n,\\ u(0)= & {} u(T) = 0. \end{aligned}$$

where \(\alpha \in (1/p, 1]\), \(1<p<\infty \), \(0 = t_0<t_1< t_2< \cdots< t_n < t_{n+1} = T\), \(f:[0,T]\times \mathbb {R} \rightarrow \mathbb {R}\) and \(I_j : \mathbb {R} \rightarrow \mathbb {R}\), \(j = 1, \ldots , n\), are continuous functions, \(a\in C[0,T]\) and

$$\begin{aligned} \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right) \\&- {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^-\right) \right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right)= & {} \lim _{t \rightarrow t_j^+} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j^-)\right)= & {} \lim _{t\rightarrow t_j^-}{_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) . \end{aligned}$$

By using variational methods and critical point theory, we give some criteria to guarantee that the above-mentioned impulsive problems have at least one weak solution and a sequences of weak solutions.

     

Multiple soliton solutions for a quasilinear Schrödinger equation

Using Morse theory, truncation arguments and an abstract critical point theorem, we obtain the existence of at least three or infinitely many nontrivial solutions for the following quasilinear Schrödinger equation in a bounded smooth domain

$$\left\{ {\begin{array}{*{20}{c}} { - {\Delta _p}u - \frac{p}{{{2^{p - 1}}}}u{\Delta _p}\left( {{u^2}} \right) = f\left( {x,u} \right)\;in\;\Omega } \\ {u = 0\;on\;\partial \Omega .} \end{array}} \right.$$

(0.1)

Our main results can be viewed as a partial extension of the results of Zhang et al. in [28] and Zhou and Wu in [29] concerning the the existence of solutions to (0.1) in the case of p = 2 and a recent result of Liu and Zhao in [21] two solutions are obtained for problem 0.1.

     

Some results on Artinian cofinite top local cohomology modules

Let \((R,\mathfrak {m})\) be a Noetherian local ring, I be an ideal of R, and M be a finitely generated R-module such that \({\text {H}}_I^t(M)\) is Artinian and I-cofinite, where \(t={\text {cd}}\,(I,M)\). In this paper, we give some equivalent conditions for the property

$$\begin{aligned} {\text {Ann}}\,_R\left( 0:_{{\text {H}}_I^t (M)} \mathfrak {p}\right) =\mathfrak {p}~\text {for all prime ideals }~ \mathfrak {p}\supseteq {\text {Ann}}\,_R{\text {H}}_I^t(M).(*) \end{aligned}$$

Also, we show that if \({\text {H}}_I^t(M)\) satisfies the property \((*)\), then \({\text {H}}_I^t(M)\cong {\text {H}}_{\mathfrak {m}}^t(M/N)\) for some submodule N of M with \({\text {dim}}\,(M/N)=t\).

     

$$L^{\infty }$$-norm and energy quantization for the planar Lane–Emden problem with large exponent

For any smooth bounded domain \(\Omega \subset {\mathbb {R}}^2\), we consider positive solutions to

$$\begin{aligned} \left\{ \begin{array}{lr}-\Delta u= u^p &{} \text{ in } \Omega \\ u=0 &{} \text{ on } \partial \Omega \end{array}\right. \end{aligned}$$

which satisfy the uniform energy bound

$$\begin{aligned}p\Vert \nabla u\Vert _{\infty }\le C\end{aligned}$$

for \(p>1\). We prove convergence to \(\sqrt{e}\) as \(p\rightarrow +\infty \) of the \(L^{\infty }\)-norm of any solution. We further deduce quantization of the energy to multiples of \(8\pi e\), thus completing the analysis performed in De Marchis et al. (J Fixed Point Theory Appl 19:889–916, 2017).

     

Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set

In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem

$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$

where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).

     

$${\varvec{\mathcal {}}}$$-minimaxness and local cohomology modules

Let R be a commutative Noetherian ring, \({\mathfrak {a}}\) an ideal of R, M a finitely generated R-module, and \({\mathcal {S}}\) a Serre subcategory of the category of R-modules. We introduce the concept of \({\mathcal {S}}\)-minimax R-modules and the notion of the \({\mathcal {S}}\)-finiteness dimension

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M):=\inf \lbrace f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{\mathfrak {p}}) \vert \mathfrak {p}\in {\text {Supp}}_R(M/ \mathfrak {a}M) \text { and } R/\mathfrak {p}\notin {\mathcal {S}} \rbrace \end{aligned}$$

and we will prove that: (i) If \({\text {H}}_{\mathfrak {a}}^{0}(M), \cdots ,{\text {H}}_{\mathfrak {a}}^{n-1}(M)\) are \({\mathcal {S}}\)-minimax, then the set \(\lbrace \mathfrak {p}\in {\text {Ass}}_R( {\text {H}}_{\mathfrak {a}}^{n}(M)) \vert R/\mathfrak {p}\notin {\mathcal {S}}\rbrace \) is finite. This generalizes the main results of Brodmann–Lashgari (Proc Am Math Soc 128(10):2851–2853, 2000), Quy (Proc Am Math Soc 138:1965–1968, 2010), Bahmanpour–Naghipour (Proc Math Soc 136:2359–2363, 2008), Asadollahi–Naghipour (Commun Algebra 43:953–958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860–4872, 2015). (ii) If \({\mathcal {S}}\) satisfies the condition \(C_{\mathfrak {a}}\), then

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M)= \inf \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\hbox {-}minimax\rbrace . \end{aligned}$$

This is a formulation of Faltings’ Local-global principle for the \({\mathcal {S}}\)-minimax local cohomology modules. (iii) \( \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\text {-minimax} \rbrace = \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not in } {\mathcal {S}} \rbrace \).

     

The effect of the domain topology on the number of solutions of fractional Laplace problems

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^n\) is an open bounded set with continuous boundary, \(n>2s\) with \(s\in (0,1),(-\Delta )^{s}\) is the fractional Laplacian operator, \(\mu \) is a positive real parameter, \(q\in [2, 2^*_s)\) and \(2^*_s=2n/(n-2s)\) is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of \(\Omega \). Precisely, we show that the problem has at least \(cat_{\Omega }(\Omega )\) nontrivial solutions, provided that \(q=2\) and \(n\geqslant 4s\) or \(q\in (2, 2^*_s)\) and \(n>2s(q+2)/q\), extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

     

Universal inequalities for eigenvalues of a system of elliptic equations of the drifting Laplacian

Let \(\Omega \) be a bounded domain in a n-dimensional Euclidean space \(\mathbb {R}^{n}\). We study eigenvalues of an eigenvalue problem of a system of elliptic equations of the drifting Laplacian

$$\begin{aligned} \left\{ \begin{array}{ll} \mathbb {L_{\phi }}\mathbf{{u}} + \alpha (\nabla (\mathrm {div}{} \mathbf{{u}}) - \nabla \phi \mathrm {div}{} \mathbf{{u}})= -\bar{\sigma }\mathbf{{u}}, &{} \hbox {in} \,\Omega ; \\ \mathbf{{u}}|_{\,\partial \Omega }=0. \end{array} \right. \end{aligned}$$

Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, a universal inequality for lower order eigenvalues of the problem is also derived. Finally, we prove an universal inequality type Ashbaugh and Benguria for the drifting Laplacian on Riemannian manifold immersed in an unit sphere or a projective space.

     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

Existence and Multiplicity of Solutions for Fractional Elliptic Problems with Discontinuous Nonlinearities

We consider the following fractional elliptic problem:

$$\begin{aligned} (P)\left\{ \begin{array}{ll} (-\Delta )^s u = f(u) H(u-\mu )&{} \quad \text{ in } \ \Omega ,\\ u =0 &{}\quad \text{ on } \ \mathbb{{R}}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$

where \((-\Delta )^s, s\in (0,1)\) is the fractional Laplacian, \(\Omega \) is a bounded domain of \(\mathbb{{R}}^n,(n\ge 2s)\) with smooth boundary \(\partial \Omega ,\) H is the Heaviside step function, f is a given function and \(\mu \) is a positive real parameter. The problem (P) can be considered as simplified version of some models arising in different contexts. We employ variational techniques to study the existence and multiplicity of positive solutions of problem (P).

     

Two nontrivial solutions of boundary-value problems for semilinear Δ<Subscript><Emphasis Type="Italic">γ</Emphasis></Subscript>-differential equations

In this paper, we study the existence of multiple solutions for the boundary-value problem

$${\Delta _\gamma }u + f\left( {x,u} \right) = 0in\Omega ,u = 0on\partial \Omega ,$$

where Ω is a bounded domain with smooth boundary in R ^N (N ≥ 2) and Δ _γ is the subelliptic operator of the type

$${\Delta _\gamma }u = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}u} \right)} ,{\partial _{{x_j}}}u = \frac{{\partial u}}{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \ldots ,{\gamma _N}} \right).$$

We use the three critical point theorem.

     

Linear Volterra equations and integrated solution families

We consider the linear Volterra equation $${\text{(VE;}}A{\text{,}}a{\text{)}}u{\text{(}}t{\text{) = }}x {\text{ + }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}Au{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$ HereA is an unbounded closed linear operator in a Banach spaceX anda is a scalar valued function. We study the theory of solution families which are not necessarily exponentially bounded and also, as their generalizations, consider the notion ofn-times integrated solution families for (VE;A, a). These families are characterized in terms of the associated Volterra integral equation $${\text{(VE;}}A{\text{,}}a{\text{)}}_n u{\text{(}}t{\text{) = }}\frac{{t^n }}{{n!}}x {\text{ + }}A{\text{ }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}u{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$ The results are applied to additive and multiplicative perturbation theorems and adjoint problems.     

Monotonicity of solutions for some nonlocal elliptic problems in half-spaces

In this paper we consider classical solutions u of the semilinear fractional problem \((-\Delta )^s u = f(u)\) in \({\mathbb {R}}^N_+\) with \(u=0\) in \({\mathbb {R}}^N {\setminus } {\mathbb {R}}^N_+\), where \((-\Delta )^s\), \(0<s<1\), stands for the fractional laplacian, \(N\ge 2\), \({\mathbb {R}}^N_+=\{x=(x',x_N)\in {\mathbb {R}}^N{:}\ x_N>0\}\) is the half-space and \(f\in C^1\) is a given function. With no additional restriction on the function f, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in \({\mathbb {R}}^N_+\) and verify

$$\begin{aligned} \frac{\partial u}{\partial x_N}>0 \quad \hbox {in } {\mathbb {R}}^N_+. \end{aligned}$$

This is in contrast with previously known results for the local case \(s=1\), where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when \(f(0)<0\).

     

A Faber–Krahn Inequality for Solutions of Schrödinger’s Equation on Riemannian Manifolds

We consider a bounded open set with smooth boundary \(\Omega \subset M\) in a Riemannian manifold (M, g), and suppose that there exists a non-trivial function \(u\in C({\overline{\Omega }})\) solving the problem

$$\begin{aligned} -\Delta u=V(x)u, \,\, \text{ in }\,\,\Omega , \end{aligned}$$

in the distributional sense, with \(V\in L^\infty (\Omega )\), where \(u\equiv 0\) on \(\partial \Omega .\) We prove a sharp inequality involving \(||V||_{L^{\infty }(\Omega )}\) and the first eigenvalue of the Laplacian on geodesic balls in simply connected spaces with constant curvature, which slightly generalises the well-known Faber–Krahn isoperimetric inequality. Moreover, in a Riemannian manifold which is not necessarily simply connected, we obtain a lower bound for \(||V||_{L^{\infty }(\Omega )}\) in terms of its isoperimetric or Cheeger constant. As an application, we show that if \(\Omega \) is a domain on a m-dimensional minimal submanifold of \({\mathbb {R}}^n\) which lies in a ball of radius R, then

$$\begin{aligned} ||V||_{L^{\infty }(\Omega )}\ge \left( \frac{m}{2R}\right) ^{2}. \end{aligned}$$