Multiplicity Of Weak Solutions Research Articles

Mixed local and nonlocal elliptic equation with singular and critical Choquard nonlinearity

In this article, we study a class of elliptic problems involving both local and nonlocal operators with different orders and a singular nonlinearity in combination with critical Hartree type nonlinearity (see problem ( P λ ) below). Using variational methods together with the critical point theory of nonsmooth analysis, we show the existence, the regularity and the multiplicity of weak solutions with respect to the parameter λ.

Existence of Multiple Weak Solutions to a Discrete Fractional Boundary Value Problem

The existence of at least three weak solutions to a discrete fractional boundary value problem containing a p-Laplacian operator and subject to perturbations is proved using variational methods. Some applications of the main results are presented. The results obtained generalize some recent results on both discrete fractional boundary value problems and p-Laplacian boundary value problems. Examples illustrating the results are given.

Multiple weak solutions for pi(x)‐Kirchhoff‐type quasilinear elliptic systems

The existence of infinitely many weak solutions for a class of quasilinear elliptic ‐Kirchhoff system is investigated by the symmetric mountain pass theorem. Furthermore, for the case where , I introduce a positive quantity to find a mountain pass weak solution and to consider the sublinear case under large growth conditions at infinity and zero.

Multiplicity results for p(x)-biharmonic equations with nonlinear boundary conditions

In this paper, we are interested in the existence of multiple weak solutions of the following fourth-order nonlinear elliptic problem with a -biharmonic operator where is a bounded domain, is the operator of fourth order called the -biharmonic operator, and , are non-negative weight functions with compact support in Ω. Our analysis mainly relies on variational arguments and some recent theory on the generalized Lebesgue–Sobolev spaces and .

Multiplicity of Solutions for Quasilinear Differential Models Generated by Instantaneous and Non-Instantaneous Impulses

This paper aims to deal with the multiplicity of weak solutions for quasilinear differential models generated by instantaneous and non-instantaneous impulses. By establishing the new variational structure and overcoming the influence of impulsive effects brought by the quasilinear term, some new results are acquired via the gene property, which extends and enriches some previous results. Moreover, an example is given to illustrate the conclusion of the main results.

Existence and multiplicity of solutions to N-Kirchhoff equations with critical exponential growth and a perturbation term

The aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: where Ω is a smooth bounded domain in and . Secondly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: where and . We assume that f and g have critical exponential growth at infinity. To establish our existence results, we use the mountain pass theorem, Ekeland variational principle and Moser–Trudinger inequality.

DIRICHLET TYPE PROBLEM WITH P(.)-TRIHARMONIC OPERATOR

In this study, we consider the existence of multiple weak solutions to a class of Dirichlet type problem involving p(.)-triharmonic {∆_p(.)^3 u=λ|u|^(q(.)-2) u in Ω, u=Δu=Δ^2 u=0 on ∂Ω, under some suitable conditions.

Existence and multiplicity of solutions for a class of damped-like fractional differential system

In this paper, we obtain some results about the existence and multiplicity of weak solutions for a class of damped-like fractional differential system with a parameter $\lambda$. When the nonlinear term is subquadratic only near the origin, we obtain that system has a ground state weak solution $u_\lambda$ if $\lambda$ is in some given interval, and when the nonlinear term is also even near the origin, then for each $\lambda>0$, system has infinitely many weak solutions $\{u_n^\lambda\}$ with $\|u_n^\lambda\|\to 0$ as $n\to \infty$. We mainly use Ekeland's variational principle and a variant of Clark's theorem together with a cut-off technique to prove our results.

Existence and Multiplicity of Weak Solutions for a Neumann Elliptic Problem with -Laplacian

AbstractWe are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic-Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to Bonanno and Marano, we guarantee the existence of at least three weak solutions for the problem in a special case.

Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

In this paper, we prove the existence of multiple weak solutions for a nonlinear nonlocal elliptic partial differential equation involving a singularity and a power nonlinearity, which is given as (−Δp)su=λuγ+uq;u>0 in Ω with zero Dirichlet boundary condition. Here, Ω is an open bounded domain in RN with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, and p−1<q≤ps*=NpN−ps. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some α ∈ (0, s], the weak solution to the problem is in C1,α(Ω¯).

Multiplicity of weak solutions to a fourth-order elliptic equation with sign-changing weight function

ABSTRACT In this paper, a fourth-order elliptic equation with a sign-changing weight function is investigated. By the modified logarithmic Sobolev inequality and Nehari manifold it is shown that the problem admits at least two nontrivial weak solutions, which shows how the sign-changing weight function affect the existence and multiplicity of weak solutions to the problem.

Multiple positive solutions for perturbed nonlinear fractional differential system with two control parameters

In this paper, we study a class of perturbed nonlinear fractional differential systems with two control parameters. Under some appropriate conditions on the primitive function of nonlinear terms F_{u} and F_{v}, we prove the multiplicity of weak solutions by using the variational method and Ricceri’s critical points theorems.

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

Large-Scale Image Geo-Localization Using Dominant Sets.

This paper presents a new approach for the challenging problem of geo-localization using image matching in a structured database of city-wide reference images with known GPS coordinates. We cast the geo-localization as a clustering problem of local image features. Akin to existing approaches to the problem, our framework builds on low-level features which allow local matching between images. For each local feature in the query image, we find its approximate nearest neighbors in the reference set. Next, we cluster the features from reference images using Dominant Set clustering, which affords several advantages over existing approaches. First, it permits variable number of nodes in the cluster, which we use to dynamically select the number of nearest neighbors for each query feature based on its discrimination value. Second, this approach is several orders of magnitude faster than existing approaches. Thus, we obtain multiple clusters (different local maximizers) and obtain a robust final solution to the problem using multiple weak solutions through constrained Dominant Set clustering on global image features, where we enforce the constraint that the query image must be included in the cluster. This second level of clustering also bypasses heuristic approaches to voting and selecting the reference image that matches to the query. We evaluate the proposed framework on an existing dataset of 102k street view images as well as a new larger dataset of 300k images, and show that it outperforms the state-of-the-art by 20 and 7 percent, respectively, on the two datasets.

Existence and multiplicity of solutions for fractional Choquard equations

We consider the following fractional order Choquard equation (−△)α∕2u(x)+(λV(x)−β)u=(|x|−μ∗|u|2μ∗)|u|2μ∗−2u,x∈Rn,with the nonlinearity in the critical growth, where α∈(0,2), n≥3, λ,β∈R+ and 2μ∗=(2n−μ)∕(n−α). Using the variational method, we establish the existence and multiplicity of weak solutions.

Existence and multiplicity of weak solutions for a nonlinear impulsive (q,p)-Laplacian dynamical system

In this paper, we investigate the existence and multiplicity of nontrivial weak solutions for a class of nonlinear impulsive (q,p)-Laplacian dynamical systems. The key contributions of this paper lie in (i) Exploiting the least action principle, we deduce that the system we are interested in has at least one weak solution if the potential function has sub-(q,p) growth or (q,p) growth; (ii) Employing a critical point theorem due to Ding (Nonlinear Anal. 25(11):1095-1113, 1995), we derive that the system involved has infinitely many weak solutions provided that the potential function is even.

WEAK SOLUTIONS FOR THE HAMILTONIAN BIFURCATION PROBLEM

We get a theorem which shows the multiple weak solutions for the bifurcation problem of the superquadratic nonlinear Hamiltonian system. We obtain this result by using the variational method, the critical point theory in terms of the S1-invariant functions and the S1-invariant linear subspaces. p(t) = −λq(t) − Hq(t,p(t),q(t)), u q(t) = λp(t) + Hp(t,p(t),q(t)), where p, q ∈ R n . Let z = (p,q) and J be the standard symplectic structure on R 2n , i.e.,

Weak solutions for a fractional equation in with Kirchhoff terms

We explore the multiplicity of weak solutions to a nonlocal equation in governed by a fractional power of the p-Laplacian combined with a Kirchhoff-type term. The nonlinearity involves two real parameters, one of which is of perturbing nature, and can overpower the critical fractional exponent. Our arguments rely upon the variational apparatus; the lack of compactness of the Sobolev embeddings is circumvented by the use of convenient weight functions.

On a class of superlinear (p,q)-Laplacian type equations on [formula omitted

Starting from a new sum decomposition of W1,p(RN)∩W1,q(RN) and using a variational approach, we investigate the existence of multiple weak solutions of a (p,q)-Laplacian equation on RN, for 1<q<p<N, with a sign-changing potential and a Carathéodory reaction term satisfying the celebrated Ambrosetti–Rabinowitz condition. Our assumptions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p-Laplacian.

Ordinary ([formula omitted])-Laplacian systems with mixed boundary value conditions

In this paper we prove the existence of multiple weak solutions for an ordinary mixed boundary value system with (p1,…,pm)-Laplacian by using recent results of critical points.