p-Laplacian Problems Research Articles

Infinitely many radial solutions for a p-Laplacian problem p-superlinear at the origin

We prove the existence of infinitely many radial solutions for a p-Laplacian Dirichlet problem which is p-superlinear at the origin. The main tool that we use is the shooting method. We extend for more general nonlinearities the results of J. Iaia in [J. Iaia, Radial solutions to a p-Laplacian Dirichlet problem, Appl. Anal. 58 (1995) 335–350]. Previous developments require a behavior of the nonlinearity at zero and infinity, while our main result only needs a condition of the nonlinearity at zero.

Infinitely many solutions of a second-order p-Laplacian problem with impulsive condition

Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a p-Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the p-Laplacian impulsive problem.

Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities

By variational methods, we prove the existence of a sign-changing solution for the p-Laplacian equation under Dirichlet boundary condition with jumping nonlinearity having relation to the Fučík spectrum of p-Laplacian. We also provide the multiple existence results for the p-Laplacian problems.

Solutions to semilinear p-Laplacian Dirichlet problem in population dynamics

In this article, we study a semilinear p-Laplacian Dirichlet problem arising in population dynamics. We obtain the Morse critical groups at zero. The results show that the energy functional of the problem is trivial. As a consequence, the existence and bifurcation of the nontrivial solutions to the problem are established.

Lyapunov Inequalities for One-Dimensional p-Laplacian Problems with a Singular Weight Function

We estimate Lyapunov inequalities for a single equation, a cycled system and a coupled system of one-dimensional -Laplacian problems with weight functions having stronger singularities than .

Non-linear eigenvalue problems for p-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.

Existence of multiple positive solutions for nth-order p-Laplacian m-point singular boundary value problems

In this paper, by using fixed point theorem, we prove the existence of multiple positive solutions for a class of nth-order p-Laplacian m-point singular boundary value problem. The interesting point is that the nonlinear term f explicitly involves the each-order derivative of variable u(t).

Multiple solutions to p-Laplacian problems with concave nonlinearities

In this paper we study the p-Laplacian type elliptic problems with concave nonlinearities. Using some asymptotic behavior of f at zero and infinity, three nontrivial solutions are established.

Nonresonance conditions for generalised ϕ-Laplacian problems with jumping nonlinearities

We consider the boundary value problem (0.1) − ψ ( x , u ( x ) , u ′ ( x ) ) ′ = f ( x , u ( x ) , u ′ ( x ) ) , a.e. x ∈ ( 0 , 1 ) , (0.2) c 00 u ( 0 ) = c 01 u ′ ( 0 ) , c 10 u ( 1 ) = c 11 u ′ ( 1 ) , where | c j 0 | + | c j 1 | > 0 , for each j = 0 , 1 , and ψ , f : [ 0 , 1 ] × R 2 → R are Carathéodory functions, with suitable additional properties. The differential operator generated by the left-hand side of (0.1), together with the boundary conditions (0.2), is a generalisation of the usual p-Laplacian, and also of the so-called ϕ-Laplacian (which corresponds to ψ ( x , s , t ) = ϕ ( t ) , with ϕ an odd, increasing homeomorphism). For the p-Laplacian problem (and more particularly, the semilinear case p = 2 ), ‘nonresonance conditions’ which ensure the solvability of the problem (0.1), (0.2), have been obtained in terms of either eigenvalues (for non-jumping f) or the Fučík spectrum or half-eigenvalues (for jumping f) of the p-Laplacian. In this paper, under suitable growth conditions on ψ and f, we extend these conditions to the general problem (0.1), (0.2).

Exact number of pseudo-symmetric positive solutions for a p-Laplacian three-point boundary value problems and their applications

In this paper, the exact number of pseudo-symmetric positive solutions is obtained for a class of three-point boundary value problems with one-dimensional p-Laplacian. The interesting point is that the nonlinearity f is general form: f(u)=λg(u)+h(u). Meanwhile, some properties of the solutions are given in details. The arguments are based upon a quadrature method.

Existence of constant sign solutions for the p-Laplacian problems in the resonant case with respect to Fučík spectrum

We consider the following the p-Laplacian equation in a bounded domain Ω: { −Δpu=f(x,u)in Ω,u=0on ∂Ω. We treat the case of nonlinearity term f satisfying the following conditions f(x,t)={ a0t+p−1−b0t−p−1+o(|t|p−1)at 0,at+p−1−bt−p−1+o(|t|p−1)at ∞, for constants a0,b0,a and b. We prove the existence of a positive solution or a negative solution in the case of (a0−λ1)(a−λ1)=0 or (b0−λ1)(b−λ1)=0 respectively, where λ1 is the first eigenvalue of −Δp.

A generalized Fucik type eigenvalue problem for p-Laplacian

In this paper we study the generalized Fucik type eigenvalue for the boundary value problem of one dimensional p Laplace type differential equations � ('(u 0 )) 0 = (u), T 1. We obtain a explicit characterization of Fucik spectrum (�,�,�,µ), i.e., for which the (*) has a nontrivial solution.

Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions

We study multiplicity of solutions for a quasilinear ellipticproblem related to the $p$-Laplacian operator. Our assumptions relyon the first eigenvalue depending on a weight function. We treatboth resonant and non-resonant cases.

Existence of solutions to three-point boundary value problem for one-dimensional p-Laplacian

In this paper, we consider the existence of solutions to three-point boundary value problem for one-dimensional p-Laplacian at resonance and non-resonance. Some known results are improved under some sign and growth conditions. The proof is based on the Brouwer degree theory.

Positive solutions for nonlinear periodic problems

We consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a nonsmooth potential. Using the degree map for multivalued perturbations of (S)+-operators and the spectrum of a weighted eigenvalue problem for the scalar periodic p-Laplacian, we prove the existence of a strictly positive solution.

Existence of radial solutions for an asymptotically linear p-Laplacian problem

We prove that an asymptotically linear Dirichlet problem which involves the p-Laplacian operator has multiple radial solutions when the nonlinearity has a positive zero and the range of the ‘ p-derivative’ of the nonlinearity includes at least the first j radial eigenvalues of the p-Laplacian operator. The main tools that we use are a uniqueness result for the p-Laplacian operator and bifurcation theory.

On a Class of Singular Sublinear p-Laplacian Problems

We establish existence results for positive solutions of the boundary value problem$¥left¥{ ¥begin{array}{l} (q(t)¥phi (u^{¥prime }))^{¥prime }+¥lambda r(t)f(u)=0,¥;t¥in (a,b) ¥¥ u(a)=u(b)=0,¥end{array}¥right.$where φ(x) = |x|p - 2 x, p > 1, q ∈ C[a,b], r ∈ L1 (a,b), f is allowed to have negative values and become infinite at 0, and λ is a positive parameter.

Successive iteration and positive symmetric solutions for some Sturm-Liouville-like four-point p-Laplacian boundary value problems

This paper deals with the existence of positive solutions for some Sturm-Liouville-like four-point p-Laplacian boundary value problems. Our approach is based on the fixed point theorem and the monotone iterative technique. Without the assumption of the existence of lower and upper solution, we obtain not only the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.

The problem of Dirichlet for evolution one-dimensional p-Laplacian with nonlinear source

In the present paper we consider the Dirichlet problem for one-dimensional p-Laplacian with nonlinear source. We obtain new a priori estimates of a solution and of the gradient of a solution and formulate conditions guaranteeing the global solvability of this problem. Our consideration includes singular case as well.

Positive solutions of multiparameter semipositone p-Laplacian problems

We obtain multiple positive solutions of multiparameter semipositone p-Laplacian problems using the sub- and supersolution method and the mountain pass lemma.