Pointwise Remez inequality

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $$[-1,1]$$ [ - 1 , 1 ] if they are bounded by 1 on a subset of $$[-1,1]$$ [ - 1 , 1 ] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded by 1 on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik–Widom bounds for the extremal value, thereby providing a complete asymptotic solution to the Andrievskii problem.

Lane-Emden equations perturbed by nonhomogeneous potential in the super critical case

Our purpose of this paper is to study positive solutions of Lane-Emden equation − Δ u = V u p i n R N ∖ { 0 } $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$ (0.1) perturbed by a non-homogeneous potential V when p ∈ [ p c , N + 2 N − 2 ) , $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ where pc is the Joseph-Ludgren exponent. When p ∈ ( N N − 2 , p c ) , $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk of −Δ u = up in ℝ N ∖ {0} by authors in [9]. While the fast decaying solution wk is unstable for p ∈ ( p c , N + 2 N − 2 ) , $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \end{array}$ so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of − Δ u = u N + 2 N − 2 $\begin{array}{} -{\it\Delta} u = u^{\frac{N+2}{N-2}} \end{array}$ in ℝ N and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.

Explicit iteration and unique solution for $ \phi $-Hilfer type fractional Langevin equations

<abstract><p>This paper proves that the monotone iterative method is an effective method to find the approximate solution of fractional nonlinear Langevin equation involving $ \phi $-Hilfer fractional derivative with multi-point boundary conditions. First, we apply a approach based on the properties of the Mittag-Leffler function to derive the formula of explicit solutions for the proposed problem. Next, by using the fixed point technique and some properties of Mittag-Leffler functions, we establish the sufficient conditions of existence of a unique solution for the considered problem. Moreover, we discuss the lower and upper explicit monotone iterative sequences that converge to the extremal solution by using the monotone iterative method. Finally, we construct a pertinent example that includes some graphics to show the applicability of our results.</p></abstract>

Extremal solution and Liouville theorem for anisotropic elliptic equations

<p style='text-indent:20px;'>We study the quasilinear Dirichlet boundary problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^{N} $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula>) is a bounded domain, and the operator <inline-formula><tex-math id="M4">\begin{document}$ Q $\end{document}</tex-math></inline-formula>, known as Finsler-Laplacian or anisotropic Laplacian, is defined by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here, <inline-formula><tex-math id="M5">\begin{document}$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}}(\xi) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ F: \mathbb{R}^{N}\rightarrow [0, +\infty) $\end{document}</tex-math></inline-formula> is a convex function of <inline-formula><tex-math id="M7">\begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $\end{document}</tex-math></inline-formula>, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if <inline-formula><tex-math id="M8">\begin{document}$ N\leq9 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We also concern the Hénon type anisotropic Liouville equation, </p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M9">\begin{document}$ \alpha&gt;-2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ F^{0} $\end{document}</tex-math></inline-formula> is the support function of <inline-formula><tex-math id="M12">\begin{document}$ K: = \{x\in\mathbb{R}^{N}:F(x)&lt;1\} $\end{document}</tex-math></inline-formula>. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for <inline-formula><tex-math id="M13">\begin{document}$ 2\leq N&lt;10+4\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ 3\leq N&lt;10+4\alpha^{-} $\end{document}</tex-math></inline-formula> respectively, where <inline-formula><tex-math id="M15">\begin{document}$ \alpha^{-} = \min\{\alpha, 0\} $\end{document}</tex-math></inline-formula>.</p>

Existence and Extremal Solution of Boundary Value Problem for Nonlinear Hybrid Fractional Differential Equation in Banach Algebras

In this work we study the existence and extremal solution for the boundary value problem of the nonlinear hybrid fractional differential equation by using hybrid fixed point theorem in Banach Algebra due to Dhage’s theorem.

Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction

We consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.

THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.

Rigidity and trace properties of divergence-measure vector fields

We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where {\varphi(t)} is a non-negative convex function vanishing only at {t=0}. We show that this property is always satisfied in dimension {n=2}, while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when {\varphi(t)=ct^{2}}) in dimension {n\geq 4}. The validity of the quadratic rigidity, which we prove in dimension {n=2}, implies the existence of the trace of a divergence-measure vector field ξ on an {\mathcal{H}^{1}}-rectifiable set S, as soon as its weak normal trace {[\xi\cdot\nu_{S}]} is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

Sharp estimates of semistable radial solutions of k-Hessian equations

We consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

Regularity of the extremal solutions associated with elliptic systems

We examine the elliptic system given by$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill &amp; {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$where λ, γ are positive parameters, Ω is a smooth bounded domain in ℝNandfis aC2positive, nondecreasing and convex function in [0, ∞) such thatf(t)/t→ ∞ ast→ ∞. Assuming$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$we show that the extremal solution (u*,v*) associated with the above system is smooth provided thatN&lt; (2α*(2 − τ+) + 2τ+)/(τ+)max{1, τ+}, where α*&gt; 1 denotes the largest root of the second-order polynomial$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$As a consequence,u*,v* ∈L∞(Ω) forN&lt; 5. Moreover, if τ−= τ+, thenu*,v* ∈L∞(Ω) forN&lt; 10.