A nonlinear noncooperative elliptic system is shown to have a positivity preserving property. That is, there exists a uniform positive constant such that, whenever the noncooperative part is bounded by this constant, positivity of the source term implies that the solution is positive. The model operator is the p-laplacian with 1 < p < ∞ on a one-dimensional domain. The source term appears in on...

This work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear differential inequalities of the form A(u) + |u|q−1u ≤ A(v) + |v|q−1v on Rn, where n ≥ 1, q > 0, and the operator A(w) belongs to a class of the so-called α-monotone operators. Typical examples of such operators are the p-Laplacian and its well-known modifications for 1 < p ≤ 2. The re...

Some asymptotic properties of principal solutions of the halflinear differential equation (*) (a(t)Φ(x′))′ + b(t)Φ(x) = 0 , Φ(u) = |u|u, p > 1, is the p-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (1),...

The calculus of generalized p-forms,−1 p n, is developed further. On an n-dimensional manifold with metric, M, a (Hodge) star operator, inner product, co-differential and Laplacian operators are introduced. The inner product and Lie derivative with respect to vector fields on M are defined. Applications are made to Hamiltonian mechanics, field theories and Einstein’s vacuum equations. PACS numb...

We consider a Dirichlet problem driven by weighted (p,2)-Laplacian with reaction which is resonant both at \(\pm\infty\) and zero (double resonance). prove multiplicity theorem producing three nontrivial smooth solutions sign information ordered. In the appendix we develop spectral properties of r-Laplace differential operator.

In this work, we obtain global solutions for nonlinear inequalities of p-Laplacian type in noncylindrical domains, the unilateral problem with strong dissipation $$ u'' -\Delta _pu-\Delta u'-f\geq 0\quad\text{in }Q_0,$$ where \(\Delta _p\) is operator \(2\leq p&lt;\infty\), and \(Q_0\) domain. Our proof based on a penalty argument by J. L. Lions Faedo-Galerkin approximations

We introduce three representation formulas for the fractional $p$-Laplace operator in whole range of parameters $0<s<1$ and $1<p<\infty$. Note that $p\ne 2$ this a nonlinear operator. The first is based on splitting procedure combines renormalized nonlinearity with linear heat semigroup. second adapts to Caffarelli-Silvestre extension technique. third one corresponding version Balakrishnan form...

&lt;abstract&gt;&lt;p&gt;This work presents some improvements on related papers that investigate certain overdetermined problems associated to elliptic quasilinear operators. Our model operator is the $ p $-Laplacian. Under suitable structural conditions, and assuming a solution exists, we show domain of problem ball centered at origin. Furthermore discuss convenient form comparison principle f...

In this paper, we study the multiplicity and concentration of positive solutions for following (p, q)-Laplacian problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u _{q} +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb {R}}^{N}, \\ u\in W^{1, p}({\mathbb {R}}^{N})\cap q}({\mathbb {R}}^{N}), \quad u>0 \end{array} \right. \end{aligned}$$ where...