Some borderline cases of nonlinear parabolic equations with irregular data

In this paper, we prove existence of solutions for nonlinear parabolic equations whose model is

$$u' - {\rm div} \, (|\nabla u|^{p-2}\nabla u) = f \quad {\rm on} \, \Omega \times (0,T),$$

with homogeneous Cauchy–Dirichlet boundary conditions, where \({1 < p < 2}\). Here f belongs to L ¹ or to L ^m , with m “small.”

     

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.

     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

Generalized <Emphasis Type="Italic">n</Emphasis>-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Let n ≥ 2 and let Ω ? ? ⁿ be an open set. We prove the boundedness of weak solutions to the problem

$$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}}{{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u}{{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$

where ? is a Young function such that the space W ₀ ¹ L ^Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L ^Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ? ⁿ .

     

Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure

We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of

$${\int}_{\Omega} f(x,u,\nabla u)\,dx $$

with f subject to the general structural conditions

$$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$

where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, g ∈ L ^t (Ω) for some t > n/p ^?, and the lower weak Harnack estimate is proved under the stronger assumption that b, g ∈ L ^∞(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, g ∈ L ^∞(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Hölder-continuous.

     

On the boundary regularity of suitable weak solutions to the Navier-Stokes equations

We consider suitable weak solutions to an incompressible viscous Newtonian fluid governed by the Navier-Stokes equations in the half space \({\mathbb {R}^3_+}\). Our main result is a direct proof of the partial regularity up to the flat boundary based on a new decay estimate, which implies the regularity in the cylinder \({Q_\rho ^+(x_0, t_0)}\) provided

$\limsup_{R\to 0}\frac {1} {R}\int\limits_{Q_R^+(x_0, t_0)} |{\rm rot}\,\mathbf u|^2 dxdt \,\leq\, \varepsilon _0$

with ε ₀ sufficiently small. In addition, we get a new condition for the local regularity beyond Serrin’s class which involves the L ²-norm of ?u and the L ^3/2-norm of the pressure.

     

Existence and multiplicity of solutions of quasilinear equations with convex or nonconvex reaction term

We give existence, nonexistence and multiplicity results of nonnegative solutions for Dirichlet problems of the form

$ - {\Delta_p}v = \lambda f(x){\left( {1 + g(v)} \right)^{p - 1}}\quad {\text{in}}\ \Omega,\quad u = 0\quad {\text{on}}\ \partial \Omega, $

where Δ _p is the p-Laplacian (p > 1), g is nondecreasing, superlinear, and possibly convex, λ > 0, and f ∈ L ¹ (Ω), f ≥ 0. New information on the extremal solutions is given. Equations with measure data are also considered.

     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.     

A modulation invariant Carleson embedding theorem outside local <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

Yen Do and Christoph Thiele developed a theory of Carleson embeddings in outer L_p spaces for the wave packet transform

$${F_\phi }(f)(u,t,\eta ) = \int {f(x){e^{i\eta (u - x)}}\phi \left( {\frac{{u - x}}{t}} \right)} \frac{{dx}}{t},(u,t,\eta ) \in R \times (0,\infty ) \times R$$

of functions f ∈ L_p(R) in the range 2 ≤ p ≤ ∞, referred to as local L². In this article, we formulate a suitable extension of this theory to exponents 1 < p < 2, answering a question posed by Do and Thiele. The proof of our main embedding theorem involves a refined multi-frequency Calderón-Zygmund decomposition in the vein of work by Di Plinio and Thiele and by Nazarov, Oberlin, and Thiele. We apply our embedding theorem to recover the full known range of L^p estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.

     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Homoclinic solutions for a class of second order discrete Hamiltonian systems

Consider the second order discrete Hamiltonian systems Δ2u(n-1)-L(n)u(n) + ▽W (n, u(n)) = f(n),where n ∈ Z, u ∈ RN and W : Z × RN → R and f : Z → RN are not necessarily periodic in n. Under some comparatively general assumptions on L, W and f , we establish results on the existence of homoclinic orbits. The obtained results successfully generalize those for the scalar case.     

Estimates for the Torsion Function and Sobolev Constants

Let Ω be an open set in Euclidean space, and let u : Ω → ?^?+? be the expected lifetime of Brownian motion in Ω. It is shown that if u?∈?L ^p (Ω) for some p?∈?[1, ?∞?) then (i) u?∈?L ^q (Ω) for all q?∈?[p,?∞?], and (ii) \({trace}\left(e^{t\Delta_{\Omega}}\right)<\infty\) for all t?>?0, where ??Δ_Ω is the Dirichlet Laplacian acting in L ²(Ω). Pointwise bounds are obtained for u in terms of the first Dirichlet eigenfunction for Ω, assuming that the spectrum of ??Δ_Ω is discrete. It is shown that if Ω is open, bounded and connected in the plane and \(\partial\Omega\) has an interior wedge with opening angle α at vertex v then the first Dirichlet eigenfunction and u are comparable near v if and only if α?≥?π/2. Two sided estimates are obtained for the Sobolev constant

$ C_p(\Omega):= \inf\left\{\Vert \nabla u \Vert_2^2: u \in C_0^{\infty}(\Omega),\ \Vert u\Vert_p = 1\right\}, $

where 0?p?Ω satisfies a strong Hardy inequality, and the distance to the boundary function δ?∈?L ^2p/(2???p)(Ω).

     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Existence of Infinitely Many Solutions for Δ<Subscript>γ</Subscript>-Laplace Problems

In this article, we study the existence of infinitelymany solutions for the boundary–value problem

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

, where Ω is a bounded domain with smooth boundary in ? ^N (N ≥ 2) and Δ_γ is a subelliptic operator of the form

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

. Our main tools are the local linking and the symmetric mountain pass theorem in critical point theory.

     

Spectrum and the trace formula for a two-dimensional Schrödinger operator in a homogeneous magnetic field

In the space L ₂(?²), we consider the operator

$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$

. We study the spectrum of H and, for V ∈ C ₀ ² (?²), prove the trace formula

$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $

where c ₀ = π ^?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ _k ⁽ⁱ⁾ are the eigenvalues of H.

     

On the regularity of scalar <Emphasis Type="Italic">p</Emphasis>-Harmonic functions in <Emphasis Type="Italic">n</Emphasis> dimensions with 2 ≤ <Emphasis Type="Italic">p</Emphasis> < <Emphasis Type="Italic">n</Emphasis>

Using Tilli’s technique [Cal Var 25(3):395–401, 2006], we shall give a new proof of the regularity of the local minima of the functional

$J\left( u\right) =\int\limits_{\Omega } \left\vert \partial u\right\vert^{p}\,dx$

with Ω a domain of class C ^{0, 1} in \({\mathbb{R}^{n}}\) and 2 ≤ p < n.

     

Dynamic control for non-linear systems with delay

We consider some class of non-linear systems of the form

$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$

where A(·) ∈ ? ^{n × n} , A _i (·) ∈ ? ^{n × n} , b(·) ∈ ? ⁿ , whose coefficients are arbitrary uniformly bounded functionals.

A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation

$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$

where ρ(·) ∈ ?¹, m(·) ∈ ? ⁿ , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.

     

Uniqueness of a positive radially symmetric Solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball

We consider the Dirichlet problem

$u_\Gamma = 0$

for the nonlinear differential equation

$\Delta u + \left| x \right|^m \left| u \right|^p = 0, x \in S,$

with constant m ≥ 0 and p > 1 in the unit ball S = {x ∈ R ⁿ : |x| < 1}(n ≥ 3) with the boundary Γ. We prove that with p ≤ m+n/n?2 this problem has a unique positive radially symmetric solution.

     

Thick clusters for the radially symmetric nonlinear Schrödinger equation

This article is devoted to the study of radially symmetric solutions to the nonlinear Schrödinger equation

$\varepsilon^2 \Delta u - V(r)u + |u|^{p-1}u = 0\, {\rm in} B,\quad \frac{\partial u}{\partial n} = 0\, {\rm on}\,{\partial}B,$

where B is a ball in \({\mathbb{R}}^N\) , 1 <  p <  (N +  2)/(N ? 2), N ≥ 3 and the potential V is radially symmetric. We construct positive clustering solutions in an annulus having O(1/?) critical points, as well as sign changing solutions with O(1/?) zeroes concentrating near zero.