Let Ω ⊂ R, n > 2, be a bounded connected domain of the class C for some θ ∈ (0, 1]. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem u ∈ W L(Ω), − div ( Φ(|∇u|) ∇u |∇u| ) + V (x)Φ(|u|) u |u| = f(x, u) + μh(x) in Ω, ∂u...

In two recent papers, Facchinei 7] and Facchinei and Kanzow 8] have shown that for a continuously diierentiable P 0-function f , the nonlinear complementarity problem NCP(f ") corresponding to the regularization f " (x) := f (x) + "x has a unique solution for every " > 0, that dist (x("); SOL(f)) ! 0 as " ! 0 when the solution set SOL(f) of NCP(f) is nonempty and bounded, and NCP(f) is stable i...

In this paper, we prove the existence of nontrivial nonnegative solutions to a class of elliptic equations and systems which do not satisfy the Ambrosetti– Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth at ∞. The known results without the AR condition in the literature only involve nonlinear terms of polynomial growth. ...

and Applied Analysis 3 DNLS equation is one of the most important inherently discrete models, which models many phenomena in various areas of applications see 2–4 and reference therein . For example, in nonlinear optics, DNLS equation appears as a model of infinite wave guide arrays. In the past decade, the existence and properties of mobile discrete solitons/breathers in DNLS equations have be...

Latent variable mixture models are a powerful tool for exploring the structure in large datasets. A common challenge for interpreting such models is a desire to impose sparsity, the natural assumption that each data point only contains few latent features. Since mixture distributions are constrained in their L1 norm, typical sparsity techniques based onL1 regularization become toothless, and co...

and Applied Analysis 3 where a ∈ [0, π), b ∈ C([0, 1], [0, +∞)) and c, γ > 0. It is easy to verify that all conditions ofTheorem 4 are satisfied. So, BVP (1)with the nonlinear term (10) has at least two solutions, one positive and the other negative. Reference [7, Theorem 3.3] can only guarantee a nonzero solution for this example. Theorem 7. Assume that (H1)–(H5) hold. Then, BVP (1) has at lea...

We derive lower bounds on the power of breather solutions ψn(t) = e −iΩtφn, Ω > 0 of a Discrete Nonlinear Schrödinger Equation with cubic or higher order nonlinearity and site-dependent anharmonic parameter, supplemented with Dirichlet boundary conditions. For the case of a defocusing DNLS, one of the lower bounds depends not only on the dimension of the lattice, the lattice spacing, and the fr...