Volume 7, Issue 3

In this paper we study the asymptotic behavior of the Steklov eigenvalues of the p- Laplacian. We show the existence of lower and upper bounds of a Weyl-type expansion of the function N(λ) which counts the number of eigenvalues less than or equal to λ, and we derive from them asymptotic bounds for the eigenvalues.

We are interested in the study of existence, uniqueness and regularity of solutions for nonlinear elliptic problems whose model is the linear boundary value problem where 0 ∊ Ω. We will prove that such results strongly depend on the size of the lower order term, that is on the constant B.

We find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I 0 to the space Symm g (R) of unordered g-tuples of points on R, and the nonlinear parallelogram is the image through the restricted generalized Abel map of the moving poles P i (x, t) (i = 1, . . . , g) of the Weyl m-functions m ± (x, t, λ) of the dynamical Sturm-Liouville family of equations L x (φ) = −φ″ + φ = λτ x (y)φ, where τ is the translation flow. It turns out that the choice of a particular stationary initial condition u g (x, t 0 ) completely determines the solution u(x, t) of all the equations in the hierarchy, as functions of the poles P i (x, t) of the Weyl m-functions corresponding to the family L x of Sturm-Liouville operators, with density function y(x, t) = u xx (x, t)/2 − 2u(x, t). For every t∊ℝ, the maps x ↦ y(x, t) lie in an isospectral class of the associated family of Sturm-Liouville equations L x (φ), and are completely determined by assigning spectral parameters and initial conditions for the poles P i (x, t).

This work is devoted to the nonexistence of positive solutions for polyharmonic systems (−Δ) m u = f(u, v), (−Δ) m v = g(u, v). Byusing the method of moving plane combined with integral inequalities and Hardy’s inequality, we prove some new Liouville type theorems for the above semilinear polyharmonic systems in ℝ N .

We study the existence and stability of standing wave for the Schrödinger-Poisson-Slater equation in three dimensional space. Let p be the exponent of the nonlinear term. Then we first show that standing wave exists for 1 < p < 5. Next, we show that when 1 < p < 7/3 and p ≠ 2, standing wave is stable for some ω > 0. We also show that when 7/3 < p < 5, standing wave is unstable for some ω > 0. Furthermore, we investigate the case of p = 2. We prove these results by using variational methods.

We study an eigenvalue problem for functions in ℝ N and we find sufficient conditions for the existence of the fundamental eigenvalue. This result can be applied to the study of the orbital stability of the standing waves of the nonlinear Schrödinger equation.

In this paper, we develop a method to compute critical groups at degenerate critical points under more general conditions. The abstract results are used to study the existence and multiplicity of nontrivial solutions for nonlinear differential equations with resonance both at infinity and at zero.

We establish new results concerning existence and the behavior at infinity of solutions for the singular nonlinear elliptic equation −Δu = ρa(x)u −α + λb(x)u γ in ℝ N , where a, b are nonnegative Hölder continuous functions with a + b > 0, α,γ > 0 are constants and ρ, γ are positive parameters. We emphasize that powers u γ with γ ≤ 1 were treated earlier by a number of authors. Our results address mainly terms u γ , where γ is any number bigger than one which we name strongly nonlinear. Penalty arguments, variational principles and an approximation procedure will be exploited.

We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure of L log L-entropy type holds. The approach relies on an a priori dimension-independent L 2 -estimate, valid for a wider class of systems including also some classical Lotka-Volterra systems, and which provides an L 1 -bound on the nonlinearities, at least for not too degenerate diffusions. In the more degenerate case, some global existence may be stated with the use of a weaker notion of renormalized solution with defect measure, arising in the theory of kinetic equations.