Search results for "-Laplacian"
showing 10 items of 103 documents
A nonlinear eigenvalue problem for the periodic scalar p-Laplacian
2014
We study a parametric nonlinear periodic problem driven by the scalar $p$-Laplacian. We show that if $\hat \lambda_1 >0$ is the first eigenvalue of the periodic scalar $p$-Laplacian and $\lambda> \hat \lambda_1$, then the problem has at least three nontrivial solutions one positive, one negative and the third nodal. Our approach is variational together with suitable truncation, perturbation and comparison techniques.
Multiplicity of positive solutions for a degenerate nonlocal problem with p-Laplacian
2021
Abstract We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the L q -norm of the solution and the p-Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their L q -norms.
The behavior of solutions of a parametric weighted (p, q)-laplacian equation
2021
<abstract><p>We study the behavior of solutions for the parametric equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda &gt;0, $\end{document} </tex-math></disp-formula></p> <p>under Dirichlet condition, where $ \Omega \subseteq \mathbb{R}^N $ is a bounded domain with a $ C^2 $-boundary $ \partial \Omega $, $ a_1, a_2 \in L^\infty(\Omega) $ with $ a_1(z), a_2(z) &gt; 0 $ for a.a. $ z \in \Omega $, $ p, q \in (1, \infty) $ and $ \Delta_{p}^{a_1}, \Delta_{q}^{a_2} $ are weighted …
Positive solutions of discrete boundary value problems with the (p,q)-Laplacian operator
2017
We consider a discrete Dirichlet boundary value problem of equations with the (p,q)-Laplacian operator in the principal part and prove the existence of at least two positive solutions. The assumptions on the reaction term ensure that the Euler-Lagrange functional, corresponding to the problem, satisfies an abstract two critical points result.
\( L^{1} \) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions
2007
Abstract In this paper we study the questions of existence and uniqueness of weak and entropy solutions for equations of type − div a ( x , D u ) + γ ( u ) ∋ ϕ , posed in an open bounded subset Ω of R N , with nonlinear boundary conditions of the form a ( x , D u ) ⋅ η + β ( u ) ∋ ψ . The nonlinear elliptic operator div a ( x , D u ) is modeled on the p-Laplacian operator Δ p ( u ) = div ( | D u | p − 2 D u ) , with p > 1 , γ and β are maximal monotone graphs in R 2 such that 0 ∈ γ ( 0 ) and 0 ∈ β ( 0 ) , and the data ϕ ∈ L 1 ( Ω ) and ψ ∈ L 1 ( ∂ Ω ) .
Weak solutions to Dirichlet boundary value problem driven by p(x)-Laplacian-like operator
2017
We prove the existence of weak solutions to the Dirichlet boundary value problem for equations involving the $p(x)$-Laplacian-like operator in the principal part, with reaction term satisfying a sub-critical growth condition. We establish the existence of at least one nontrivial weak solution and three weak solutions, by using variational methods and critical point theory.
Weak solution for Neumann (p,q)-Laplacian problem on Riemannian manifold
2019
We prove the existence of a nontrivial solution for a nonlinear (p, q)-Laplacian problem with Neumann boundary condition, on a non compact Riemannian manifold. The idea is to reduce the problem in variational form, which means to consider the critical points of the corresponding Euler-Lagrange functional in an Orlicz-Sobolev space. (C) 2019 Elsevier Inc. All rights reserved.
Multiplicity theorems for the Dirichlet problem involving the p-Laplacian
2003
Multiplicity theorems for the Dirichlet problem involving the p-Laplacian were proved using variational approach. It was shown that there existed an open interval and a positive real number, and each problem admits at least three weak solutions. Results on the existence of at least three weak solutions for the Dirichlet problems were established.
Multiple solutions for a Neumann-type differential inclusion problem involving the p(.)-Laplacian
2012
Using a multiple critical points theorem for locally Lipschitz continuous functionals, we establish the existence of at least three distinct solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian.
Landesman-Lazer type (p, q)-equations with Neumann condition
2020
We consider a Neumann problem driven by the (p, q)-Laplacian under the Landesman-Lazer type condition. Using the classical saddle point theorem and other classical results of the calculus of variations, we show that the problem has at least one nontrivial weak solution.