Equations Of p-Laplacian Type Research Articles

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Holder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

Calderón-Zygmund estimates for nonlinear equations of differential forms with BMO coefficients

We obtain Lq-regularity estimates for weak solutions to p-Laplacian type equations of differential forms. In particular, we prove local Calderón-Zygmund type estimates for equations with discontinuous coefficients satisfying the bounded mean oscillation (BMO) condition.

Global bound on the gradient of solutions to p-Laplace type equations with mixed data

Global bound on the gradient of solutions to p-Laplace type equations with mixed data

P-Laplacian Type Equations Via Mountain Pass Theorem in Cerami Sense

p-Laplacian Type Equations Via Mountain Pass Theorem in Cerami Sense

Existence and regularity for nonlinear parabolic double obstacle problems

We study nonlinear parabolic equations of p-Laplacian type with irregular double obstacles to establish the existence and an optimal global Calderón–Zygmund theory.

A global second-order Sobolev regularity for p-Laplacian type equations with variable coefficients in bounded domains

A global second-order Sobolev regularity for p-Laplacian type equations with variable coefficients in bounded domains

Global existence and blow-up for wave equation of p-Laplacian type

Global existence and blow-up for wave equation of p-Laplacian type

On Nirenberg’s Problem for Compact Perturbations of Expansive Operator and an Application to p-Laplacian Type Equation with Nonmonotone Convection Function

On Nirenberg’s Problem for Compact Perturbations of Expansive Operator and an Application to p-Laplacian Type Equation with Nonmonotone Convection Function

Global existence and finite time blowup for a mixed pseudo-parabolic [formula omitted]-Laplacian type equation

This paper concerns the initial–boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then prove the existence, uniqueness and decay estimate of global solutions and the blowup phenomena of solutions with subcritical initial energy. Next, we extend parallelly these results to the critical initial energy. Lastly, the existence, uniqueness and asymptotic behavior of global solutions with supercritical initial energy are proved by further analyzing the properties of ω-limits of solutions.

Uniqueness of extremals for some sharp Poincaré-Sobolev constants

We study the sharp constant for the embedding ofW01,p(Ω)W^{1,p}_0(\Omega )intoLq(Ω)L^q(\Omega ), in the case2>p>q2>p>q. We prove that for smooth connected sets, whenq>pq>pandqqis sufficiently close topp, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side.The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions ofp−p-Laplace–type equations by L. Damascelli and B. Sciunzi.

On the p-Laplacian type equation with logarithmic nonlinearity: existence, decay and blow up

This work is deal with a problem of wave equation with p-Laplacian, strong damping and logarithmic source terms under initial-boundary conditions. The global existence of weak solution was proved for related to the equation. Global existence results of solutions are obtained using the potential well method, Galerkin method and compactness approach corresponding to the logarithmic source term. Besides, we established the energy functional decaying polynomially to zero as the time goes to infinity due to Nakao?s inequality and some precise priori estimates on logarithmic nonlinearity. For suitable conditions we proved the finite time blow up results of solutions. The proof is based on the concavity method, perturbation energy method and differential-integral inequality technique. Additionally, under suitable assumptions on initial data, the infinite time blow up result is investigated with negative initial energy.

Nonlinear Singular Elliptic Equations of p-Laplace Type with Superlinear Growth in the Gradient

Nonlinear Singular Elliptic Equations of p-Laplace Type with Superlinear Growth in the Gradient

Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

In this article, we apply blow-up analysis to study pointwise a priori estimates for some p-Laplacian equations based on Liouville type theorems. With newly developed analysis techniques, we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function, i.e., the solution of Δpu = 0, x ∈ Ω. We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplacian equation −Δpu = |u|λ − 1u, x ∈ Ω, which are then extended to the equation with general right hand term f(x, u) with certain asymptotic properties. In addition, pointwise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplacian equation, in a case of p = 2, are also discussed.

Interior regularity results for inhomogeneous anisotropic quasilinear equations

We consider inhomogeneous p-Laplace type equations of the form $$-\textrm{div}\left( a(\nabla u)\right) =f$$ in a possibly anisotropic setting. Under general assumptions on the source term f, we obtain quantitative Sobolev regularity results for the stress field $$a(\nabla u)$$ and weighted $$L^2$$ estimates for the Hessian of the solution. As far as we know, our results are new or refine the ones available in literature also when restricted to the Euclidean setting.

Global existence, nonexistence, and decay of solutions for a wave equation of p-Laplacian type with weak and p-Laplacian damping, nonlinear boundary delay and source terms

In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases p > 2 and p = 2. To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term.

Gradient estimates for singular parabolic p-Laplace type equations with measure data

We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic p-Laplace equation $$u_t-\Delta _p u=\mu $$ with $$p\in (1,2)$$ . The case when $$p\in \big (2-\frac{1}{n+1},2\big )$$ were studied in Kuusi and Mingione (Ann Sc Norm Super Pisa Cl Sci 5 12(4):755–822, 2013). In this paper, we extend the results in Kuusi and Mingione (2013) to the open case when $$p\in \big (\frac{2n}{n+1},2-\frac{1}{n+1}\big ]$$ if $$n\ge 2$$ and $$p\in (\frac{5}{4}, \frac{3}{2}]$$ if $$n=1$$ . More specifically, in a more singular range of p as above, we establish pointwise gradient estimates via linear parabolic Riesz potential and gradient continuity results via certain assumptions on parabolic Riesz potential.

Regularity estimates for singular parabolic measure data problems with sharp growth

We prove global gradient estimates for parabolic p-Laplace type equations with measure data, whose model isut−div(|Du|p−2Du)=μinΩ×(0,T)⊂Rn×R,where μ is a signed Radon measure with finite total mass. We consider the singular case2nn+1<p≤2−1n+1and give possibly minimal conditions on the nonlinearity and the boundary of Ω, which guarantee the regularity results for such measure data problems.

Unilateral Problem for a Viscoelastic Beam Equation Type p-Laplacian with Strong Damping and Logarithmic Source

In this manuscript, we investigate the unilateral problem for a viscoelastic beam equation of p-Laplacian type. The competition of the strong damping versus the logarithmic source term is considered. We use the potential well theory. Taking into account the initial data is in the stability set created by the Nehari surface, we prove the existence and uniqueness of global solutions by using the penalization method and Faedo-Galerkin’s approximation.

On fractional p-Laplacian type equations with general nonlinearities

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: \begin{equation*} \left\{ \begin{array}{l} (-\Delta)_p^s u= f(x,u) \quad \hfill \textrm{in} \ \Omega,\\ \quad u=0 \ \hfill \textrm{in} \ R^N \setminus \Omega, \end{array} \right.\\ \end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, $\Omega$ is an open bounded subset of $R^N$ with Lipschitz boundary and $f:\Omega \times R \to R$ is a generic Carath\'eodory function satisfying either a $p-$sublinear or a $p-$superlinear growth condition.

Large Deviation Principle for McKean–Vlasov Quasilinear Stochastic Evolution Equations

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.