Search results for "Laplace operator"
showing 10 items of 101 documents
Three solutions for parametric problems with nonhomogeneous (a,2)-type differential operators and reaction terms sublinear at zero
2019
Abstract We consider parametric Dirichlet problems driven by the sum of a Laplacian and a nonhomogeneous differential operator ( ( a , 2 ) -type equation) and with a reaction term which exhibits arbitrary polynomial growth and a nonlinear dependence on the parameter. We prove the existence of three distinct nontrivial smooth solutions for small values of the parameter, providing sign information for them: one is positive, one is negative and the third one is nodal.
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 …
A note on homoclinic solutions of (p,q)-Laplacian difference equations
2019
We prove the existence of at least two positive homoclinic solutions for a discrete boundary value problem of equations driven by the (p,q) -Laplace operator. The properties of the nonlinearity ensure that the energy functional, corresponding to the problem, satisfies a mountain pass geometry and a Palais–Smale compactness condition.
Invariant Markov semigroups on quantum homogeneous spaces
2019
Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected coideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting on expected coideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators …
Ordinary (p1,…,pm)-Laplacian systems with mixed boundary value conditions
2016
Abstract In this paper we prove the existence of multiple weak solutions for an ordinary mixed boundary value system with ( p 1 , … , p m )-Laplacian by using recent results of critical points.
Multiplicity of solutions to a nonlinear boundary value problem of concave–convex type
2015
Abstract Problem (P) { − Δ p u + | u | p − 2 u = | u | r − 1 u x ∈ Ω | ∇ u | p − 2 ∂ u ∂ ν = λ | u | s − 1 u x ∈ ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, ν is the unit outward normal at ∂ Ω , Δ p is the p -Laplacian operator and λ > 0 is a parameter, was studied in Sabina de Lis (2011) and Sabina de Lis and Segura de Leon (in press). Among other features, it was shown there that when exponents lie in the regime 1 s p r , a minimal positive solution exists if 0 λ ≤ Λ , for a certain finite Λ , while no positive solutions exist in the complementary range λ > Λ . Furthermore, in the radially symmetric case a second positive solution exists for λ varying in the same full range ( 0 , Λ ) …
Principal eigenvalue of a very badly degenerate operator and applications
2007
Abstract In this paper, we define and investigate the properties of the principal eigenvalue of the singular infinity Laplace operator Δ ∞ u = ( D 2 u D u | D u | ) ⋅ D u | D u | . This operator arises from the optimal Lipschitz extension problem and it plays the same fundamental role in the calculus of variations of L ∞ functionals as the usual Laplacian does in the calculus of variations of L 2 functionals. Our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of Berestycki, Nirenberg and Varadhan [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operator…
\( L^{1} \) 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 ) .
Fixed point methods and accretivity for perturbed nonlinear equations in Banach spaces
2020
Abstract In this paper we use fixed point theorems to guarantee the existence of solutions for inclusions of the form A u + λ u + F u ∋ g , where A is a quasi-m-accretive operator defined in a Banach space, λ > 0 , and the nonlinear perturbation F satisfies some suitable conditions. We apply the obtained results, among other things, to guarantee the existence of solutions of boundary value problems of the type − Δ ρ ( u ( x ) ) + λ u ( x ) + F u ( x ) = g ( x ) , x ∈ Ω , and ρ ( u ) = 0 on ∂Ω, where the Laplace operator Δ should be understood in the sense of distributions over Ω and to study the existence and uniqueness of solution for a nonlinear integro-differential equation posed in L 1 …
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.