p-Laplacian Operator Research Articles

Existence of two weak solutions for some singular elliptic problems

In this paper, we establish the existence of at least two distinct weak solutions for some singular elliptic problems involving a p-Laplace operator, subject to Dirichlet boundary conditions in a smooth bounded domain in \(\mathbb {R}^N.\) A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least two distinct non-trivial weak solutions.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR PROBLEMS WITH WEIGHTED P-LAPLACIAN AND P-BIHARMONIC OPERATORS Vol. 3(2) July 2015, No. 16, pp. 215-226 A. C. CAVALHEIRO

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR PROBLEMS WITH WEIGHTED P-LAPLACIAN AND P-BIHARMONIC OPERATORS Vol. 3(2) July 2015, No. 16, pp. 215-226 A. C. CAVALHEIRO

Homogenization for the p-Laplace operator in perforated media with nonlinear restrictions on the boundary of the perforations: A critical Case

Homogenization for the p-Laplace operator in perforated media with nonlinear restrictions on the boundary of the perforations: A critical Case

On the Dependence on p of the Variational Eigenvalues of the p-Laplace Operator

We study the behavior of the variational eigenvalues of the p-Laplace operator, with homogeneous Dirichlet boundary condition, when p is varying. After introducing an auxiliary problem, we characterize the continuity answering, in particular, a question raised in Lindqvist (Potential Anal. 2(3), 199–218 1993).

Existence and uniqueness results for q-fractional difference equations with p-Laplacian operators

In this paper, we consider the following two-point boundary value problem for q-fractional p-Laplace difference equations. New results on the existence and uniqueness of solutions for q-fractional boundary value problem are obtained. These results extend the corresponding ones of ordinary differential equations of integer order. Finally, an example is presented to illustrate the validity and practicability of our main results.

Existence of solutions of fractional boundary value problems with p-Laplacian operator

In this paper, the existence of the solutions of the fractional differential equation with p-Laplacian operator and integral conditions is discussed. By Green’s functions and the fixed point theorems, we state and prove the existence and uniqueness results of the problem. Two examples are given to illustrate the results.

An optimal mass transport approach for limits of eigenvalue problems for the fractional p-Laplacian

AbstractWe find an interpretation using optimal mass transport theory for eigenvalue problems obtained as limits of the eigenvalue problems for the fractional p-Laplacian operators as p → +∞. We deal both with Dirichlet and Neumann boundary conditions.

Optimal regularity for the obstacle problem for the p-Laplacian

In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such thatdiv(|∇u|p−2∇u)≤0,u≥ϕ,in B1, with ϕ∈C1,1(B1) and given boundary datum on ∂B1. We prove that the solution is uniformly C1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L∞. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C1,1p−1 in the spatial variables.

Critical Group Estimates for Nonregular Critical Points of Functionals Associated With Quasilinear Elliptic Equations

We consider a class of quasilinear elliptic equations whose principal part includes the p-area and the p-Laplace operators, when p lies in a suitable left neighborhood of 2. For the critical points of the associated functional, we provide estimates of the corresponding critical groups, under assumptions that do not guarantee any further regularity of the critical point.

Positive solutions for a class of quasilinear problems with critical growth in ℝN

In this paper, we study the existence, multiplicity and concentration of positive solutions for a class of quasilinear problemswhere —Δp is the p-Laplacian operator for is a small parameter, f(u) is a superlinear and subcritical nonlinearity that is continuous in u. Using a variational method, we first prove that for sufficiently small ε > 0 the system has a positive ground state solution uε with some concentration phenomena as ε → 0. Then, by the minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials. Finally, we obtain some sufficient conditions for the non-existence of ground state solutions.

Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger–Kirchhoff-type equations

In this paper, we are concerned with the following Schrödinger–Kirchhoff-type problem:(P){−(a+b∫RN|∇u|pdx)p−1Δpu+λV(x)|u|p−2u=f(x,u),x∈RN,u∈W1,p(RN), where a>0, b≥0 are constants, Δpu:=div(|∇u|p−2∇u) is the p-Laplacian operator with p≥2, V(x) is the potential function satisfying some conditions which may not guarantee the compactness of the corresponding Sobolev embedding. By using the variational methods, we prove the existence and multiplicity of nontrivial solutions for problem (P).

Multiple positive solutions of nonlinear third-order boundary value problems with integral boundary conditions on time scales

In this paper, we consider a class of nonlinear third-order boundary value problems with integral boundary conditions on time scales. By applying a generalization of the Leggett-Williams fixed point theorem, we establish the existence of at least three positive solutions. The discussed problem involves both an increasing and positive homomorphism, which generalizes the p-Laplacian operator. As an application, we give an example to illustrate our results.

Existence and uniqueness of positive and nondecreasing solutions for a class of fractional boundary value problems involving the p-Laplacian operator

In this article, we investigate the existence of a solution arising from the following fractional q-difference boundary value problem by using the p-Laplacian operator: $D_{q}^{\gamma}(\phi_{p}(D_{q}^{\delta}y(t)))+f(t,y(t))=0$ ( $0< t<1$ ; $0<\gamma<1$ ; $3<\delta<4$ ), $y(0)=(D_{q}y)(0)=(D_{q}^{2}y)(0) =0$ , $a_{1}(D_{q}y)(1)+a_{2}(D_{q}^{2}y)(1)=0$ , $a_{1} +\vert a_{2}\vert \neq0$ , $D_{0+}^{\gamma}y(t)|_{t=0}=0$ . We make use of such a fractional q-difference boundary value problem in order to show the existence and uniqueness of positive and nondecreasing solutions by means of a familiar fixed point theorem.

On the Lipschitz stability of inverse nodal problem for p-Laplacian Schrödinger equation with energy dependent potential

In this study, we consider reconstruction and stability issues of an inverse nodal problem for a p-Laplacian Schrodinger equation with energy dependent potential. We solve Lipschitz stability of the inverse nodal problem for this p-Laplacian operator. Furthermore, we show that the space of all potential functions q is homeomorphic to the partition set of all asymptotically equivalent nodal sequences induced by an equivalence relation.

Multiple Solutions of Nonlinear Fractional Differential Equations with p-Laplacian Operator and Nonlinear Boundary Conditions

Abstract In this paper, we deal with multiple solutions of fractional differential equations with p-Laplacian operator and nonlinear boundary conditions. By applying the Amann theorem and the method of upper and lower solutions, we obtain some new results on the multiple solutions. An example is given to illustrate our results.

Multiplicity of Solutions to a Concave–Convex Problem

Abstract This paper studies two related problems. A first one, where Δpu = div(|∇u|p−2∇u) stands for the p-Laplacian operator, Ω is a ball in ℝN, ν stands for the outer unit normal and λ &gt; 0 is a parameter. Exponents are supposed to satisfy 1 &lt; q &lt; p &lt; r ≤ p*, p* = N p/(N − p) if 1 &lt; p &lt; N, p* = ∞ otherwise. The existence of Λ &gt; 0 is shown so that (P) does not admit positive solutions if λ &gt; Λ, a minimal positive solution exists when 0 &lt; λ ≤ Λ and most importantly, a further second positive solution arises if 0 &lt; λ &lt; Λ. Hence, extensions of results in [2], [3], [12] and [13] in the framework of (P) are provided. The second problem considered is the variant of (P): where Ω is a ball and 1 &lt; q &lt; p &lt; r. Features described above are shown to be also exhibited by (Q) and more importantly, it is proved that minimal solutions to (Q) develop flat patterns in the degenerate regime p &gt; 2. Finally, it should be stressed that some of the properties satisfied by (P) and (Q) hold true when Ω is a general smooth domain.

A variational approach to multiplicity results for boundary-value problems on the real line

We study the existence and multiplicity of solutions for a parametric equation driven by the p-Laplacian operator on unbounded intervals. Precisely, by using a recent local minimum theorem we prove the existence of a non-trivial non-negative solution to an equation on the real line, without assuming any asymptotic condition either at 0 or at ∞ on the nonlinear term. As a special case, we note the existence of a non-trivial solution for the problem when the nonlinear term is sublinear at 0. Moreover, under a suitable superlinear growth at ∞ on the nonlinearity we prove a multiplicity result for such a problem.

Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p &gt; 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.

Extremal solutions for p-Laplacian fractional integro-differential equation with integral conditions on infinite intervals via iterative computation

We study the extremal solutions of a class of fractional integro-differential equation with integral conditions on infinite intervals involving the p-Laplacian operator. By means of the monotone iterative technique and combining with suitable conditions, the existence of the maximal and minimal solutions to the fractional differential equation is obtained. In addition, we establish iterative schemes for approximating the solutions, which start from the known simple linear functions. Finally, an example is given to confirm our main results.

Stability for [formula omitted]-Laplace type equation in a borderline case

We study the Dirichlet problem for a p-Laplacian type operator in the setting of the Orlicz–Zygmund space ℒqlog−αℒ(Ω,RN), q>1 and α>0. More precisely, our aim is to establish under which assumptions on α>0 existence and uniqueness of the solution are assured and to prove continuity and stability of the associated nonlinear operator.