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We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent p ⁢ (⋅) -Laplacian as a forward model with the interior density data σ ⁢ | ∇ ⁡ u | q , and find out that single measurement specifies the boundary conductivity when p - q ≥ 1 , and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both p and q may depend on the spatial variable x, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of positive solutions of an equation with p-Laplacian where n ≥ 3, 1 < p < n, q > 0, Δpu = div(|∇u|p−2∇u), and K(x) is a double bounded function. We will prove the following results. When 0 < q < p − 1, the equation has no solution satisfying inf Rnu = 0 for any double bounded function K(x). When q = p − 1, the equation has a positive radial singular solution for K(x) ≡ Const.. In addition, q = p − 1 is a necessary condition such that this equation (with K(x) ≡ 1) has finite energy solutions. In addition, the existence of positive solutions are also derived for the system where u, v > 0 in Rn \ {0}, n ≥ 3, 1 < p1, p2 < n, q1, q2 > 0, and K1(x), K2(x) are double bounded functions. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the existence of radial solutions of the boundary value problem of p-Laplace equation with gradient term { − Δ p u = K ( | x | ) f ( | x | , u , | ∇ u | ) , x ∈ Ω , ∂ u ∂ n = 0 , x ∈ ∂ Ω , lim | x | → ∞ u ( x ) = 0 , $$ \textstyle\begin{cases} -\Delta_{p} u= K( \vert x \vert ) f( \vert x \vert , u, \vert \nabla u \vert ) ,\quad x\in\Omega , \\ \frac{\partial u}{\partial n}=0 ,\quad x\in\partial\Omega, \\ \lim_{ \vert x \vert \to\infty}u(x)=0 , \end{cases} $$ where Ω = { x ∈ R N : | x | > r 0 } $\Omega=\{x\in\mathbb{R}^{N}: |x|>r_{0}\}$ , N ≥ 3 $N\ge3$ , 1 < p ≤ 2 $1< p\le2$ , K : [ r 0 , ∞ ) → R + $K: [r_{0}, \infty)\to\mathbb{R}^{+}$ , and f : [ r 0 , ∞ ) × R × R + → R $f: [r_{0}, \infty)\times\mathbb{R}\times\mathbb{R}^{+}\to \mathbb{R}$ are continuous. Under certain inequality conditions of f, the existence results of radial solutions are obtained.

This paper studies the existence of periodic solutions of Duffing type p-Laplace equations (ϕp (x′)) ′ + g (x) = p (t). When g has a singularity and satisfies one-sided non-resonant condition, we prove the existence of periodic solutions by using the phase plane analysis method and the continuation lemma. [ABSTRACT FROM AUTHOR]

This paper studies the existence of radial solutions of the boundary value problem of p-Laplace equation with gradient term { − Δ p u = K (| x |) f (| x | , u , | ∇ u |) , x ∈ Ω , ∂ u ∂ n = 0 , x ∈ ∂ Ω , lim | x | → ∞ u (x) = 0 , where Ω = { x ∈ R N : | x | > r 0 } , N ≥ 3 , 1 < p ≤ 2 , K : [ r 0 , ∞) → R + , and f : [ r 0 , ∞) × R × R + → R are continuous. Under certain inequality conditions of f, the existence results of radial solutions are obtained. [ABSTRACT FROM AUTHOR]

We prove Gibbons' conjecture for the quasilinear elliptic equation - Δ p ⁢ u + a ⁢ (u) ⁢ | ∇ ⁡ u | q = f ⁢ (u) in ⁢ ℝ N , where N ≥ 2 , 2 ⁢ N + 2 N + 2 < p < 2 , q ≥ 1 and a and f are Lipschitz continuous functions which satisfy some relevant conditions. This conjecture states that every weak solution u ∈ C 1 ⁢ (ℝ N) of the equation with | u | ≤ 1 and lim x N → ± ∞ ⁡ u ⁢ (x ′ , x N) = ± 1 , uniformly in x ′ ∈ ℝ N - 1 , must depend only on x N and ∂ ⁡ u ∂ ⁡ x N > 0 in ℝ N . In particular, our result holds for a being non-decreasing on [ - 1 , - 1 + δ ] and on [ 1 - δ , 1 ] and f ⁢ (u) = | u | r ⁢ u ⁢ | 1 - u 2 | s ⁢ (1 - u 2) , where r , s , δ ≥ 0 . The main tool we use is an adaptation of the sliding method to the corresponding quasilinear elliptic operator. [ABSTRACT FROM AUTHOR]

Beals, Gaveau, and Greiner established a formula for the fundamental solution to the Laplace equation with drift term in Grushin-type planes. The first author and Childers expanded these results by invoking a p-Laplacetype generalization that encompasses these formulas while the authors explored a different natural generalization of the p-Laplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity. [ABSTRACT FROM AUTHOR]

In order to study the properties of P-Laplace equation in differential dynamical systems, a class of long time behavior problems of P-Laplace equation with delay term defined in homeomorphic region was discussed. Firstly, in order to overcome the difficulty caused by the time change of the region on the non-cylindrical region, the homeomorphic coordinate transformation was used to convert the non-cylindrical region into the columnar region, and a series of prior estimates were built on the columnar region. Then, the fittest results of the weak solutions of the system were obtained by the Faedo-Galerkin method. Finally, the asymptotic compactness of the system was proved by the energy method. The results show that there is a unique pullback attractor for the P-Laplace equation with delay in the noncylindrical region. [ABSTRACT FROM AUTHOR]

We consider quasilinear elliptic equations Δ p u + f (u) = 0 in the quarter-plane Ω, with zero Dirichlet data. For some general nonlinearities f, we prove the existence of a positive solution with a prescribed limiting profile. The question is motivated by the result in (Adv. Nonlinear Stud.13(1) (2013) 115–136), where the authors establish that for solutions u (x 1 , x 2) of the preceding Dirichlet problem, lim x 1 → ∞ u (x 1 , x 2) = V (x 2) , where V is a solution of the corresponding one-dimensional problem with V (+ ∞) = z and z is a root of f. Starting with such a profile V and a carefully selected z, the authors of this paper apply Perron's method in order to prove the existence of a solution u with limiting profile V. The work in this paper is similar in spirit to that in (Math. Methods Appl. Sci.39(14) (2016) 4129–4138), where the authors compare the sub and the super solutions by using arguments based on the strong maximum principle for semilinear equations. However, for the quasilinear case, such a maximum principle is lacking. This difficulty is overcome by employing a less classical weak sweeping principle that requires a careful boundary analysis. [ABSTRACT FROM AUTHOR]

We consider the discrete Raviart-Thomas mixed finite element method (dRT-MFEM) for the p-Laplace equation in the new sense of measurement. The new measurement of p-Laplace equation for 2 < p < was studied by D. J. Liu (APPL. NUMER. MATH., 152: 323-337, 2020), where the reliable error analysis for conforming and nonconforming FEM were obtained. This paper provide the reliable and efficient error analysis of dRT-MFEM for p-Laplace equation (1 < p < 2). The numerical investigation for benchmark problem demonstrates the accuracy and robustness of the proposed dRT-MFEM. [ABSTRACT FROM AUTHOR]

Let Gλ, λ > 0, be a Grushin type operator on R n × R m . In this paper, we will first establish existence and uniqueness for the p-Laplace equation for Gλ, then we prove the regularity of solutions. In order to prove our results, we will get the Harnack inequality for Gλ, which maybe has independent interest. [ABSTRACT FROM AUTHOR]

Our main interest is to demonstrate the existence of at least one positive radial solution for the p-Laplacian problem { − d i v (| ∇ u | p − 2 ∇ u) + R (x) | u | p − 2 u = f (u) − g (u) x ∈ B , u > 0 x ∈ B , u = 0 x ∈ ∂ B , where B is the unit ball centred at the origin in R n , n ≥ 3 and p ≥ 2 , via the variational method. [ABSTRACT FROM AUTHOR]

We consider a p-Laplace problem in a strip with two-constant boundary Dirichlet conditions. We show that if the width of the strip is smaller than some d0 ∈ (0, +∞], then the problem admits a unique bounded solution, which is strictly monotone.Hence this unique solution is one-dimensional symmetric and belongs to the C² class. We also show that the problem has no bounded solution in the case that d0 < +∞ and the width of the strip is larger than or equal to d0. An analogous rigidity result in the whole space was obtained recently by Esposito et al. [8] [ABSTRACT FROM AUTHOR]

This paper deals with a class of fast diffusion p-Laplace equation with logarithmic non-linearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates and some ordinary differential inequalities, we study the conditions on extinction and non-extinction of global solutions. The results of this paper extend and complete the previous studies on this equation. [ABSTRACT FROM AUTHOR]

We analyze the asymptotic behavior of a non-Newtonian Stokes system, posed in a Hele–Shaw cell, i.e. a thin three-dimensional domain which is confined between two curved surfaces and contains a cylindrical obstacle. The fluid is assumed to be of power-law type defined by the exponent 1 < p < ∞. By letting the thickness of the domain tend to zero we obtain a generalized form of the Poiseuille law, i.e. the limit velocity is a nonlinear function of the limit pressure gradient. The flow is assumed to be driven by an external pressure which is applied as a normal stress along the lateral part of the boundary. On the remaining part of the boundary we impose a no-slip condition. The two-dimensional limit problem for the pressure is a generalized form of the p ′ -Laplace equation, 1 / p + 1 / p ′ = 1 , with a coefficient called 'flow factor', which depends on the geometry as well as the power-law exponent. The boundary conditions are preserved in the limit as a Dirichlet condition for the pressure on the lateral boundary and as a Neumann condition for the pressure on the solid obstacle. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the asymptotic behavior for a class of McKean-Vlasov stochastic partial differential equations with slow and fast time-scales. Using the variational approach and classical Khasminskii time discretization, we show that the slow component strongly converges to the solution of the associated averaged equation. In particular, the corresponding convergence rates are also obtained. The main results can be applied to demonstrate the averaging principle for various McKean-Vlasov nonlinear SPDEs such as stochastic porous media type equation, stochastic p -Laplace type equation and also some McKean-Vlasov stochastic differential equations. [ABSTRACT FROM AUTHOR]

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This paper is devoted to investigating the Freidlin–Wentzell's large deviation principle for a class of McKean–Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean–Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean–Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations. [ABSTRACT FROM AUTHOR]

We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations of the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of Bieske and Childers, in that Bieske and Childers consider a generalization that focuses on the p-Laplace-type equation while we primarily concentrate on a generalization of the drift term. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of fast diffusion p-Laplace equation with singular potential in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, we get the solution of the equation exists globally. Moreover, the conditions of extinction and non-extinction are studied. The results of this paper extend and complete the previous studies on this equation. [ABSTRACT FROM AUTHOR]

We consider elliptic equations of the p-Laplace type with dual data. We obtain interior and boundary gradient estimates for their weak solutions under sharp conditions on the dual data, coefficients and boundaries of their domains. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of p-Laplace equations. Using variational methods, we prove that there are two solutions and one of these solutions is nonnegative. Using recurrence method, we prove that there are infinitely many solutions to this class of equations. [ABSTRACT FROM AUTHOR]

Abstract In this article, a functional minimum problem equivalent to the p-Laplace equation is introduced, a finite element-Newton iteration formula is established, and a well-posed condition of iterative functions satisfied is provided. According to the well-posed condition, an effective initial iterative function is presented. Using the effective particular initial function and Newton iterations with the iterative step length equal to 1, an effective particular sequence of iterative functions is obtained. With the decreasing properties of gradient modulus of subdivision finite element, it has been proved that the function sequence converges to the solution of finite element formulation of p-Laplace equation. Moreover, a discussion on local convergence rate of iterative functions is provided. In summary, the iterative method based on the effective particular initial function not only makes up the shortage of the Newton algorithm, which requires an exploratory reduction in the iterative step length, but also retains the benefit of fast convergence rate, which is verified with theoretical analysis and numerical experiments.

In this note we study solutions defined on the whole space R^N for the p-Laplace equation $$ \hbox{div}(| \nabla u|^{p-2}\nabla u)+f(u)=0. $$ Under an appropriate condition on the growth of f, which is weaker than conditions previously considered in McCoy [3] and Cuccu-Mhammed-Porru [1], we prove the non-existence of non-trivial positive solutions.

In this paper, we study the existence and the uniqueness of a positive mild solution for the following singular nonlinear problem with homogeneous Dirichlet boundary conditions: (St) ∂tu - Δpu = u -δ + f(x,u) in (0,T) × Ω =: QT, u = 0 on (0,T) × ∂Ω, u > 0 in QT, u(0,x) = u0 ≥ 0 in Ω, where Ω stands for a regular bounded domain of ℝN, Δpu is the p-Laplacian operator defined by Δpu = div(|∇u|p-2|∇u|) 1 < p < ∞, δ > 0 and T > 0. The nonlinear term f : Ω × ℝ → ℝ is a bounded below Carathéodory function and nonincreasing with respect to the second variable (for a.e. x ∈ Ω). We prove for any initial positive data u0 ∈ 𝒟(A)¯L∞$\overline{{\mathcal {D}}(A)}^{L^\infty }$ the existence of a mild solution to (St). Then, we deduce some stabilization results for problem (St) in L∞(Ω) when p ≥ 2. This complements some results obtained in [J. Differential Equations 252 (2012), 5042–5075] stated with the additional restriction δ < 2 + 1/(p - 1).

We consider the Dirichlet problem for the p-Laplace equation with nonlinear gradient terms. In particular, these gradient terms cannot satisfy the Bernstein—Nagumo conditions. We obtain some sufficient conditions that guarantee the existence of a global bounded radially symmetric solution without any restrictions on the growth of the gradient term. Also we present some conditions on the function simulating the mass forces, which allow us to obtain a bounded radially symmetric solution under presence of an arbitrary nonlinear source. [ABSTRACT FROM AUTHOR]

We present a proof of the local Hölder regularity of the horizontal derivatives of weak solutions to the p-Laplace equation in the Heisenberg group ℍ1 for p > 4. [ABSTRACT FROM AUTHOR]

In this paper we develop a method to solve (stochastic) evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and p-Laplace equations as special cases. [ABSTRACT FROM AUTHOR]

This paper is concerned with the blow-up of solutions to the following nonlocal p-Laplace equation: under homogeneous Neumann boundary conditions in a bounded smooth domain $${\Omega\subset\mathrm{R}^N}$$ . For all $${p > 2, q > p-1}$$ , a blow-up result for the solutions to the above equation with positive initial energy is established. This result improves a recent result by Qu and Liang (Abstr Appl Anal 3:551-552, 2013) which asserts the blow-up of solutions for $${p-1 < q\leq\frac{Np}{(N-p)_+}-1}$$ . [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of random attractors for the p-Laplace equation driven simultaneously by non-autonomous deterministic and stochastic forcing. The nonlinearity of the equation is allowed to have a polynomial growth rate of any order which may be greater than p. We further establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. In addition, we show the pathwise periodicity of random attractors when all non-autonomous deterministic forcing terms are time periodic.

In this article, a functional minimum problem equivalent to the p-Laplace equation is introduced, a finite element-Newton iteration formula is established, and a well-posed condition of iterative functions satisfied is provided. According to the well-posed condition, an effective initial iterative function is presented. Using the effective particular initial function and Newton iterations with the iterative step length equal to 1, an effective particular sequence of iterative functions is obtained. With the decreasing properties of gradient modulus of subdivision finite element, it has been proved that the function sequence converges to the solution of finite element formulation of p-Laplace equation. Moreover, a discussion on local convergence rate of iterative functions is provided. In summary, the iterative method based on the effective particular initial function not only makes up the shortage of the Newton algorithm, which requires an exploratory reduction in the iterative step length, but also retains the benefit of fast convergence rate, which is verified with theoretical analysis and numerical experiments. [ABSTRACT FROM AUTHOR]

The p-Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p-Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p-Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p-Laplace equation into the p-Dirac equation. This equation will be solved iteratively by using a fixed-point theorem. Applying operator-theoretical methods for the p-Dirac equation and p-Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We recover the gradient of a scalar conductivity defined on a smooth bounded open set in ℝd from the Dirichlet to Neumann map arising from the p-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when p ≠ 2. In the p = 2 case boundary determination plays a role in several methods for recovering the conductivity in the interior. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem for an inhomogeneous p-Laplace equation with nonlinear source in the presence of exterior mass forces.We obtain new sufficient conditions for the existence of a weak nonnegative bounded solution. The sufficient conditions are written in explicit form through the data of the problem. [ABSTRACT FROM AUTHOR]

In this short paper, the authors investigate the extinction and non-extinction of solutions to a fast diffusive p-Laplace equation with a nonlocal source and an absorption term. By applying the super-solution and sub-solution methods, instead of energy estimate methods, they give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which generalizes some previous results.

We consider Hele-Shaw flows driven by injection of a highly shear-thinning power-law fluid (of exponent n) in the absence of surface tension. We formulate the problem in terms of the streamfunction ψ, which satisfies the p-Laplacian equation ∇·(|∇ψ|p−2∇ψ) = 0 (with p = (n+1)/n) and use the method of matched asymptotic expansions in the large n (extreme-shear-thinning) limit to find an approximate solution. The results show that significant flow occurs only in (I) segments of a (single) circle centred on the injection point, whose perimeters comprise the portion of free boundary closest to the injection point and (II) an exponentially small region around the injection point and (III) a transition region to the rest of the fluid: while the flow in the latter is exponentially slow it can be characterised in detail. [ABSTRACT FROM AUTHOR]

We investigate a version of the Phragmén-Lindelöf theorem for solutions of the equation Δ∞u = 0 in unbounded convex domains. The method of proof is to consider this infinity harmonic equation as the limit of the p-harmonic equation when p tends to ∞. [ABSTRACT FROM AUTHOR]

According to a class of constrained minimization problems, the Schwartz symmetrization process and the compactness lemma of Strauss, we prove that there is a nontrivial ground state solution for a class of p-Laplace equations without the Ambrosetti-Rabinowitz condition. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with pullback attractors of the stochastic p-Laplace equation defined on the entire space . We first establish the asymptotic compactness of the equation in and then prove the existence and uniqueness of non-autonomous random attractors. This attractor is pathwise periodic if the non-autonomous deterministic forcing is time periodic. The difficulty of non-compactness of Sobolev embeddings on is overcome by the uniform smallness of solutions outside a bounded domain. [Copyright &y& Elsevier]