In particular, in the case of N-Laplacian, i.e., a x,∇u)="|∇u|N−2∇u, we obtain multiplicity of weak solutions of (0.1).
Some papers of Chang [7] and Choi and Jung [8] considered the existence and multiplicity of weak solutions for nonlinear boundary value problems with asymptotically linear term.
Second, we show the multiplicity of weak solutions to the quasilinear Choquard equation (P) via the fountain theorem to obtain the infinitely many weak solutions.
Under the condition of (1.8), [16] proved the existence and multiplicity of weak solutions for the nonuniformly nonlinear problem Δ ( a ( x, Δ u ) ) = f ( x, u ), in Ω, u = ∂ u ∂ n = 0 on ∂ Ω, (1.9).
In the past decades, many authors have considered the existence and multiplicity of weak solutions for the elliptic equations and the elliptic systems by the variational method (see [1 3] for the semilinear elliptic equations, [4, 5] for the semilinear elliptic systems, [6 9] for p-Laplacian equations, [10, 11] for p-Laplacian systems, [12 16] for ((p,q -elliptic systems,q -ellipticcesystemsin).
Qu and Tang [5] obtained the existence and multiplicity of weak solutions of problem (1.2) by using the Ekeland variational principle, the mountain pass theorem, and the saddle point theorem in critical point theory, and by applying the local linking theorem and the saddle point theorem some new existence theorems of weak solutions were obtained by Duan et al. [6].
We investigate the multiplicity of the weak solutions for the nonlinear elliptic boundary value problem.
In this paper, we investigate the existence and multiplicity of nontrivial weak solutions for a class of nonlinear impulsive ((q,p -Laplacian dynamical systems.
In Yang [26], the author derives similar results for the bi-Laplacian operator in dimension four and Yang [27] constructs the existence and multiplicity of a weak solution for the N-Laplacian elliptic equation.
In this paper, we establish some results about the existence and multiplicity of nontrivial weak solution to nonlinear elliptic equations of the p ( x ) -Laplacian type, − div ( φ ( x, ∇ u ) ) + | u | p ( x ) − 2 u = λ f ( x, u ) in R N, (B).
Ultimately, the multiplicity of individual weak proto-silencers and anti-silencers in core X and Y'44 build up "buffering" cis-elements, which suppress extreme variations in TPE.
