p-Laplacian Elliptic Equations Research Articles

Optimality conditions for optimal control of non-well-posed p-Laplacian elliptic equations

Optimality conditions for optimal control of non-well-posed p-Laplacian elliptic equations

Existence of multiple solutions for a nonhomogeneous p-Laplacian elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of multiple nontrivial solutions for nonhomogeneous p-Laplacain elliptic problems involving the critical Hardy-Sobolev exponent. The method used here is based on the Nehari manifold.

Existence results for elliptic problems with gradient terms via a priori estimates

We prove existence of nonnegative solutions of a Dirichlet problem on a bounded smooth domain of RN for a p-Laplacian elliptic equation with a convection term. Our proof is based on a priori bounds for a suitable weighted norm involving the distance function from the boundary, obtained by adapting the technique developed by Barrios et al. [4] for nonlocal elliptic problems, which is a modification of the classical scaling blow up method due to Gidas and Spruck in the celebrated paper [25]. The conclusion then follows by using topological degree.

Discretization of Optimal Control Problems Governed by p-Laplacian Elliptic Equations

In this paper, a state-constrained optimal control problem governed by p-Laplacian elliptic equations is studied. The feasible control set or the cost functional may be nonconvex, and the purpose is to obtain the convergence of a solution of the discretized control problem to an optimal control of the relaxed continuous problem.

On nonhomogeneous p-Laplacian elliptic equations involving a critical Sobolev exponent and multiple Hardy-type terms

On nonhomogeneous p-Laplacian elliptic equations involving a critical Sobolev exponent and multiple Hardy-type terms

Existence of Solutions for a Fractional and Non-Local Elliptic Operator

In this paper, we consider a fractional and p-laplacian elliptic equation. In order to study this problem, we apply the technique of Nehari manifold and fibering map, which permit treating the existence of nontrivial solutions of a fractional and nonlocal equation, satisfies the homogeneous Dirichlet boundary conditions.

Existence and multiplicity of positive solutions for p-Laplacian elliptic equations

We study a p-Laplacian elliptic equation with Hardy term and Hardy-Sobolev critical exponent, where the nonlinearity is $(p-1)$ -sublinear near zero and $(p^{\ast}(s)-1)$ -sublinear near infinity ( $p^{\ast}(s)=\frac{p(N-s)}{N-p}$ is the Hardy-Sobolev critical exponent). By using variational methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation. To the best of our knowledge, no result has been published concerning the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation.

Existence and boundary behavior of solutions to p-Laplacian elliptic equations

Under appropriate conditions on $b(x)$ and $g(u)$ , we consider the singular Dirichlet problems $-\Delta_{p} u=b(x)g(u)$ , $u>0$ , $x \in \Omega $ , $u\vert_{\partial \Omega }=0$ . These problems are shown to admit weak solutions, and we analyze their exact asymptotic behavior near the boundary. As the main tools, we use Karamata regular variation theory and the method of upper and lower solutions.

Large solutions with a power nonlinearity given by a variable exponent for [formula omitted]-Laplacian equations

In this paper we establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to p-Laplacian elliptic equations with a power nonlinearity given by a variable exponent −Δpu=b(x)uq(x), where Δpu=div(|∇u|p−2∇u) with p>1. Here we combine the localization method originated by López-Gómez (2003) with some previous radially symmetric results to get the asymptotic behavior near the boundary.

Multiple solutions for a p-Laplacian elliptic problem

We consider the following p-Laplacian elliptic equation on W 1 , p ( R N ): − Δ p u+b(|x|) | u | p − 2 u=f(|x|,u). For certain f(|x|,u), we are interested in the functional on a group invariant subspace, and we obtain the existence of infinitely many radial solutions and non-radial solutions of the equation, which extends the result of (Bartsch and Willem in J. Funct. Anal. 117:447-460, 1993) to the space W 1 , p ( R N ).

Boundary blow-up solutions of [formula omitted]-Laplacian elliptic equations with a weakly superlinear nonlinearity

This paper studies the asymptotic behavior near the boundary for boundary blow-up solutions to p-Laplacian elliptic equations −Δpu=a(x)h(u)−b(x)f(u) in a smooth bounded domain Ω of RN with N≥2, where Δpu=div(∣∇u∣p−2∇u) with p>1. We assume that f(u) does not grow like uq with q>p−1 or faster at infinity, but behaves like up−1lnαu as u→∞ for some α>p, and h(u) does not grow faster than up−1 at infinity. This case is more difficult to handle than those two cases (f(u) grows like uq with q>p−1 or faster at infinity). Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. Uniqueness of such solutions would then follow as a consequence.

Boundary blow-up solutions for p-Laplacian elliptic equations of logistic type

We establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to p-Laplacian elliptic equations of logistic type −Δpu = a(x)h(u) − b(x)f(u), where Δpu = div (|∇u|p−2∇u) with p > 1, h(u)/up−1 is non-increasing and f(u) is a function whose variation at infinity may be regular or rapid. In particular, our result regarding the blow-up rate reveals the main difference between regular variation function f and rapid variation function f.

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δpu = a(x)um−b(x)f(u) with p > 1 and 0 1.

Multiplicity of solutions for a singular p-laplacian elliptic equation

The existence of two continuous solutions for a nonlinear singular elliptic equation with natural growth in the gradient is proved for the Dirichlet problem in the unit ball centered at the origin. The first continuous solution is positive and maximal; it

Multiplicity of solutions for a noncooperative [formula omitted]-Laplacian elliptic system in [formula omitted

In this paper, we study a system of p-Laplacian elliptic equations Δ p u - | u | p - 2 u = F u ( | x | , u , v ) in R N , - Δ p v + | v | p - 2 v = F v ( | x | , u , v ) in R N , u , v ∈ W 1 , p ( R N ) . Applying Principle of Symmetric Criticality and a Limit Index Theory, under some assumptions on F u , F v , we can obtain the existence of infinitely many radial solutions and nonradial solutions for the above system, which extends some results in Li (Nonlinear Anal. TMA 25 (1995) 1371).

The existence of multiple solutions of p-laplacian elliptic equation

This paper considers the quasilinear elliptic equation (1.1)λ{-Δpu=|u|m-1u+λ|u|q-1u, x∈Ω,u∈W01,p(Ω),where -Δpu=-div(|Δu|p-2Δu), and 0<m<p-1<q<+∞, ω is a bounded domain in RN(N≥3). λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ lias at least one positive solution if λ υ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ υ (0,λ*) and q≤NpN-p-1. Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0<m<p-1<q=NpN-p-1.