Band 8, Heft 2

Let Ω be a domain in ℝ d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Qʹ given by It is assumed that Q ≥ 0 on C 0 ∞ (Ω). In a previous paper [22] we discussed relations between the absence of weak coercivity of the functional Q on C 0 ∞ (Ω) and the existence of a generalized ground state. In the present paper we study further relationships between functional-analytic properties of the functional Q and properties of positive solutions of the equation Qʹ(u) = 0.

In this paper, we consider nonlinear Dirichlet problem driven by the p-Laplacian differential operator. Using variational methods based on the critical point theory and truncation techniques, we prove the existence of at least three nontrivial smooth solutions. The hypotheses on the nonlinearity incorporate in our framework of analysis both coercive and noncoercive problems. For the semilinear problem (p = 2), using Morse theory, we show the existence of four nontrivial smooth solutions.

In this paper, under the framework of Ambrosetti [1], using mountain-pass theorem and topological degree theory, we extend the result of Ambrosetti [1] (in which H is independent of λ and C 1 ) to the equation Au + H(λ, u) = λu with H jointly continuous in (λ, u) and H(λ, ·) of class C 0, 1 . We give a description of the structure of local bifurcation from isolated eigenvalues of A: the set of bifurcation solutions at each isolated eigenvalue of A contains at least one connected branch. Applications to the existence of branching points to semilinear elliptic partial differential equations are given.

Let A : D(A) → E, D(A) ⊂ E; be an infinitesimal generator either of an analytic compact semigroup or of a contractive C 0 -semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of T- periodic solutions for the equation ẋ = Ax + f(t, x) + εg(t, x, ε) from a k-parameterized family of T-periodic solutions of the unperturbed equation corresponding to ε = 0. We show that by means of a suitable modification of the classical Mel’nikov approach we can construct a bifurcation function and to formulate the conditions for the existence of bifurcation in terms of the topological index of the bifurcation function. To do this, since the perturbation term g is only Lipschitzian we need to extend the classical Lyapunov- Schmidt reduction to the present nonsmooth case.

The main result of this work is to get the existence of infinitely many radial positive solutions to the problem -Δ p u = |▽u| q + λf(x) in Ω, u|aΩ = 0, where Ω = B 1 (0) and f is a radial positive function. Since, in general when q ≠ p, a Hopf-Cole type change can not be used, we will consider just the existence and multiplicity of positive radial solutions. The main idea is to get a relation between radial positive solutions of the above equation and a suitable quasilinear family of problems with measures data that we will make precise later.

Abelian gauge theories consist of a class of field equations which provide a model for the interaction between matter and electromagnetic fields. In this paper we analyze the existence of solitary waves for these theories. We assume that the lower order term W is positive and we prove the existence of solitary waves if the coupling between matter and electromagnetic field is small. We point out that the positiveness assumption on W implies that the energy is positive: this fact makes these theories more suitable to model physical phenomena.

We deal with the nonlinear Schrödinger equation -Δu + V(x)u = f(u) in ℝ N , where V is a (possible) sign changing potential satisfying mild assumptions and the nonlinearity f ∈ C 1 (ℝ, ℝ) is a subcritical and superlinear function. By combining variational techniques and the concentration-compactness principle we obtain a positive ground state solution and also a nodal solution. The proofs rely in localizing the infimum of the associated functional constrained to Nehari type sets.

We use the mountain-pass theorem combined with the principle of symmetric criticality to establish multiplicity of solutions for the class of quasilinear elliptic equations -Δu + V(z)u - Δ(u 2 )u = h(u) in ℝ N where N ≥ 4, the potential V : ℝ N → ℝ is positive and bounded away from zero and satisfies appropriate periodic and symmetric conditions. The nonlinear term h(u) has subcritical growth and satisfies a condition of the Ambrosetti-Rabinowitz type. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

By Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u| q , u > 0, x ∈ Ω, u| aΩ = 0, where Ω is a bounded domain with smooth boundary, λ ∈ ℝ, q ∈ [0, 2]; g ∈ C 1 ((0, ∞), (0, ∞)), is decreasing in (0, ∞) with lim s→0 + g(s) = +∞; the weight b is positive in Ω and singular on the boundary.

We study the existence of solutions for the semilinear elliptic equation where 1 < p < , Q(x) is a sign-changing function such that both the supports of Q + and Q - may have infinite measure. We show that the problem has at least one solution.

This paper deals with the instability of the ground state solitary wave solution to the Klein-Gordon-Zakharov system in three space dimensions with c ≥ 1, which is a model to describe the Langmuir turbulence in plasma. First we construct a suitable constrained variational problem and obtain the existence of the standing waves with ground state by using variational calculus and scaling argument. Then by defining invariant sets and applying some priori estimates, we prove the orbital instability of the ground state in the following sense: in each neighborhood of it, there exists a solution whose energy diverges in finite or infinite time.