Existence of Positive Solutions for Higher Order <Emphasis Type="Italic">p</Emphasis>-Laplacian Boundary Value Problems

In this paper, we consider a higher order p-Laplacian boundary value problem

$$\begin{aligned} (-1)^n[\phi _{p}(u^{(2n-2)}+k^2u^{(2n-4)})]''=f(t,u), ~~0\le t\le 1,\\ u^{(2i)}(0)=0=u^{(2i)}(1), ~~ 0\le i \le n-1, \end{aligned}$$

where \(n\ge 1\) and \(k\in (0, \frac{\pi }{2})\) is a constant. By applying fixed point index theory, we derive sufficient conditions for the existence of positive solutions to the boundary value problem.

     

Some Optimization Problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian Type Equations

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional over some admissible class of loads f where u is the (unique) solution to the problem −Δ _p u+|u| ^p−2 u=0 in Ω with |∇ u| ^p−2 u _ν =f on ∂Ω. Supported by Universidad de Buenos Aires under grant X078, by ANPCyT PICT No. 2006-290 and CONICET (Argentina) PIP 5478/1438. J. Fernández Bonder is a member of CONICET. Leandro M. Del Pezzo is a fellow of CONICET.     

The Positive Solutions for Integral Boundary Value Problem of Fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian Equation with Mixed Derivatives

In this paper, we consider a class of integral boundary value problems of fractional p-Laplacian equation, which involve both Riemann–Liouville fractional derivative and Caputo fractional derivative. By using the generalization of Leggett–Williams fixed point theorem, some new results on the existence of at least three positive solutions to the boundary value problems are obtained. Finally, some examples are presented to illustrate the extensive potential applications of our main results.     

Existence and Uniqueness of Positive Solutions of a Kind of Multi-point Boundary Value Problems for Nonlinear Fractional Differential Equations with <Emphasis Type="BoldItalic">p</Emphasis>-Laplacian Operator

In this paper, we investigate the existence and uniqueness of positive solutions of a kind of multi-point boundary value problems for nonlinear fractional differential equations with p-Laplacian operator using the Banach contraction mapping principle. Furthermore, some examples are given to illustrate our results.     

Generation and Propagation of Interfaces for <Emphasis Type="Italic">p</Emphasis>-Laplacian Equations

In this paper, we consider the generation and propagation of interfaces for p-Laplacian equations with the derivative of a bi-stable potential.     

Asymmetric critical <Emphasis Type="Italic">p</Emphasis>-Laplacian problems

We obtain nontrivial solutions for two types of asymmetric critical p-Laplacian problems with Ambrosetti–Prodi type nonlinearities in a smooth bounded domain in \({\mathbb {R}}^N,\, N \ge 2\). For \(1< p < N\), we consider an asymmetric problem involving the critical Sobolev exponent \(p^*= Np/(N - p)\). In the borderline case \(p = N\), we consider an asymmetric critical exponential nonlinearity of the Trudinger–Moser type. In the absence of a suitable direct sum decomposition, we use a linking theorem based on the \({\mathbb {Z}}_2\)-cohomological index to prove existence of solutions.     

On the Regularity of a Boundary Point for the <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)-Laplacian

The Dirichlet problem for the p(x)-Laplacian with a continuous boundary function is considered, and a sufficient condition is found for the continuity of its solution at a boundary point, assuming that the exponent p(x) satisfies the log-Hölder continuity condition at this point.     

On Existence Results for Nonlinear Fractional Differential Equations Involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian at Resonance

In this paper, we study the existence result for the nonlinear fractional differential equations with p-Laplacian operator

$$\left\{\begin{array}{ll}D_{0^+}^{\beta} \phi_p( D_{0^+}^{\alpha} u(t))=f(t,u(t),D_{0^+}^{\alpha}u(t)), \quad t\in(0,1),\\ D_{0^+}^{\alpha}u(0)=D_{0^+}^{\alpha}u(1)=0,\end{array}\right.$$

where the p-Laplacian operator is defined as \({\phi_p(s) = |s|^{p-2}s,p > 1, \,\,{\rm and}\,\, \phi_q(s) = \phi_p^{-1}(s), \frac{1}{p}+\frac{1}{q} = 1;\, 0 < \alpha, \beta < 1, 1 < \alpha + \beta < 2 \,\,{\rm and}\,\, D_{0^+}^{\alpha}, D_{0^+}^{\beta}}\) denote the Caputo fractional derivatives, and \({f : [0,1] \times \mathbb{R}^2\rightarrow \mathbb{R}}\) is continuous. Though Chen et al. have studied the same equations in their article, the proof process is not rigorous. We point out the mistakes and give a correct proof of the existence result. The innovation of this article is that we introduce a new definition to weaken the conditions of Arzela–Ascoli theorem and overcome the difficulties of the proof of compactness of the projector K _P (I ? Q)N. As applications, an example is presented to illustrate the main results.

     

Periodic Solutions for a Kind of Duffing Type <Emphasis Type="Italic">p</Emphasis>-Laplacian Neutral Equation

By means of the generalized Mawhin’s continuation theorem, we present some sufficient conditions which guarantee the existence of at least one T-periodic solution for a p-Laplacian neutral functional differential equation with multiple deviating arguments.     

Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term

Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u _k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.     

Faber-Krahn inequality for robin problems involving <Emphasis Type="Italic">p</Emphasis>-Laplacian

The eigenvalue problem for the p-Laplace operator with Robin boundary conditions is considered in this paper.A Faber-Krahn type inequality is proved.More precisely,it is shown that amongst all the domains of fixed volume,the ball has the smallest first eigenvalue.     

Non-linear eigenvalue problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian with variable domain

In this work we consider some eigenvalue problems for p-Laplacian with variable domain. Eigenvalues of this operator are taken as a functional of the domain. We calculate the first variation of this functional, using the obtained formula investigate behavior of the eigenvalues when the domain varies. Then we consider one shape optimization problem for the first eigenvalue, prove the necessary condition of optimality relatively domain, offer an algorithm for the numerical solution of this problem.     

A multiplicity theorem for <Emphasis Type="Italic">p</Emphasis>-superlinear <Emphasis Type="Italic">p</Emphasis>-Laplacian equations using critical groups and morse theory

We study a nonlinear elliptic equation driven by the Dirichlet p-Laplacian and with a Carathéodory nonlinearity. We assume that the nonlinearity exhibits a p-superlinear growth near infinity but need not satisfy the Ambrosetti–Rabinowitz condition. Using truncation techniques, minimax methods and Morse theory, we show that the problem admits at least three nontrivial solutions, two of which have constant sign (one positive, the other negative).     

A two-phase obstacle-type problem for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

We study the so-called two-phase obstacle-type problem for the p-Laplacian when p is close to 2. We introduce a new method to obtain the optimal growth of the function from branch points, i.e. two-phase points in the free boundary where the gradient vanishes. As a by-product we can locally estimate the (n − 1)-Hausdorff-measure of the free boundary for the special case when p > 2. This research project is a part of the ESF program Global. E. Lindgren has been supported by grant KAW 2005.0098 from the Knut and Alice Wallenberg foundation.     

Boundary blow-up solutions of <Emphasis Type="Italic">p</Emphasis>-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type ?Δ _p u = a(x)u ^m ?b(x)f(u) with p >  1 and 0 <  m <  p?1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

On a Harnack inequality for the elliptic (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian

A special Harnack inequality is proved for solutions of nonlinear elliptic equations of the p(x)-Laplacian type with a variable exponent p(x) that takes different values on two sides of a hyperplane dividing the domain. Examples are given showing that the classical Harnack inequality does not hold in this case.     

Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems

Abstract   The singular second-order m-point boundary value problem

Oscillation criteria for PDE with <Emphasis Type="Italic">p</Emphasis>-Laplacian involving general means

Oscillation criteria for PDE with p-Laplacian

Nonlinear commutators for the fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian and applications

We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded \(\frac{n}{s}\)-harmonic maps converge strongly outside at most finitely many points.     

Hadamard variational formulas for <Emphasis Type="Italic">p</Emphasis>-torsion and <Emphasis Type="Italic">p</Emphasis>-eigenvalue with applications

In this paper, we pose two kinds of Minkowski problems involving the p-Laplacian operator. The Hadamard variational formulas for some p-Laplacian functionals are obtained. A good application is to prove symmetry results for solutions to some overdetermined problems of p-Laplacian equations.