Existence Of Multiple Weak Solutions Research Articles

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.

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 .

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 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¯).

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.

Multiple solutions of p-biharmonic equations with Navier boundary conditions

In this article, exploiting variational methods, the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator is investigated. Moreover, a concrete example of an application is presented.

Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator

The existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the $p$--biharmonic operator is investigated. Our approach is chiefly based on critical point theory.

Existence of multiple weak solutions for asymptotically linear wave equations

We consider the problem of multiple existence of 2 π -periodic weak solutions to wave equations □ u ( x , t ) = h ( x , t , u ( x , t ) ) + f ( x , t ) of space dimension 1, where h ( x , t , ξ ) is asymptotically linear in ξ both as ξ → 0 and as | ξ | → ∞ . It is shown by variational methods that there exist at least three solutions under several conditions on h ( x , t , ξ ) if f is sufficiently small. One of the results reads as follows. Let b ≔ lim | ξ | → ∞ ∂ h / ∂ ξ ( x , t , ξ ) and assume that the convergence is uniform with respect to ( x , t ) and that b ∉ σ ( □ ) (non-resonant case). Then the following conditions guarantee the existence of at least three solutions for sufficiently small f: (a) h ( x , t , ξ ) - ξ ∂ h / ∂ ξ ( x , t , 0 ) is non-decreasing (resp. non-increasing) in ξ , and sup ( x , t , ξ ) ∂ h ∂ ξ ( x , t , ξ ) < min { λ ∈ σ ( □ ) ; b < λ } resp . inf ( x , t , ξ ) ∂ h ∂ ξ ( x , t , ξ ) > max { λ ∈ σ ( □ ) ; b > λ } . To obtain these results, we prove that a C 1 -class functional, which is bounded from above and satisfies a condition similar to the linking condition, has at least three critical points under ( WPS ) * condition.