Volume 14, Issue 2

We discuss the existence and properties of solutions for systems of Dirichlet problems involving one dimensional mean curvature operator. Our approach is based on variational methods and covers both sublinear and superlinear cases of nonlinearities. We also investigate the continuous (in some sense) dependence of solutions on functional parameters.

The paper is concerned about an improvement of Moser-Trudinger inequality involving L p norm for a bounded domain in n dimensions. Let be the first eigenvalue associated with n-Laplacian. We obtain the following strengthened Moser-Trudinger inequality with blow-up analysis for 0 ≤ α < λ̅(Ω) and 1 < p ≤ n, and the supremum is infinity for α ≥ λ̅(Ω), where and ω n−1 is the surface area of the unit ball in ℝ n . We also obtain the existence of the extremal functions for (0.2).

We consider a general boundary value problem involving operators of the form div(a(·, ∇u(·)) in which a is a Carathéodory function satisfying a p(·)-growth condition. We are interested on the weak solvability of the problem and, to this end, we start by introducing the Lebesgue and Sobolev spaces with variable exponent, together with their main properties. Then, we state and prove our main existence and uniqueness result, Theorem 3.1. The proof is based on a Weierstrass-type argument. We also consider two antiplane contact problems for nonhomogenous elastic materials of Hencky-type. The contact is frictional and it is modelled with a regularized version of Tresca’s friction law and a power-law friction, respectively. We prove that the problems cast in the abstract setting, then we use Theorem 3.1 to deduce their unique weak solvability.

In this paper we consider the Dirichlet problem with mean curvature operator in Minkowski space : where Ω ⊂ ℝ N is a smooth open bounded set and f : Ω × ℝ → ℝ is a Carathéodory function. We show that u is a solution of the above problem iff it is a critical point of the corresponding non-smooth action functional. Applications concerning nontrivial solutions for a class of such Dirichet problems depending on a parameter are provided.

We prove the existence of two nontrivial solutions for the variational inequality ∫ Ω ∇u∇(v − u) ≥ ∫ Ω f(u)(v − u) for every v belonging to some convex set, where Ω ⊂ ℝ 2 . The function f has critical exponential growth, in the sense that it behaves like exp(α 0 s 2 ) as |s| → ∞, for some α 0 > 0. We use variational methods for lower semicontinuous functionals.

In this paper, we consider the existence of solutions of the following problem where Ω ⊂ ℝ N , N ≥ 2α, is a smooth bounded domain, λ > 0, β > 0, 0 < α < 2, and is the critical exponent. We show that problem (0.1) possesses a ground state solution provided that N = 2α and λ m < λ < λ m+1 for some m ∈ℕ and β > 0 small or ℕ > (1 + √ 2)α and λ m ≤ λ < λ m+1 for some m ∈ ℕ and β > 0 small, where λ m is the eigenvalue of −Δ with Dirichlet boundary condition.

Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k 1 ξ p ≤ f (ξ) ≤ k 2 ξ p for all ξ ≥ 0 and some k 1 , k 2 > 0 and p ∈ (0, 1). We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form −Δu = m(x) f (u) in Ω, u = 0 on ∂Ω.

Consider the semilinear Schrödinger equation where f is a superlinear, subcritical nonlinearity. We mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −Δ + V. Based on the work of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009], we develop a new technique to show the boundedness of Cerami sequences and derive a new super-quadratic condition that there exists a θ 0 ∈ (0, 1) such that for the existence a “ground state solution” which minimizes the corresponding energy among all nontrivial solutions. Our result unifies and improves some known ones and the recent ones of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009] and Liu [Calc. Var. 45: 1-9, 2012].

In this paper twist symplectic maps with non-periodicity are considered, where the associated generating functions grow exponentially. Under appropriate assumptions the existence of infinitely many bounded complete orbits is established. An application to Toda-like type lattices may help us to explain our results.

In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S 1 -equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.

In this work we consider the Fučik problem for a family of weights depending on ε with Dirichlet and Neumann boundary conditions. We study the homogenization of the spectrum. We also deal with the special case of periodic homogenization and we obtain the rate of convergence of the first non-trivial curve of the spectrum.

Using an algebraic topological method and the tools of the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the the n-dimensional balls.

We consider the existence of positive bound states for the nonlinear Schrödinger equation −ε 2 Δu + V(x)u = u p , u > 0, u ∈ H 1 (ℝ N ), where N ≥ 3, 1 < p < (N + 2)/(N − 2), and V is a nonnegative potential with compact support. For arbitrary positive integer k ∈ Z+, we construct higher energy solutions with exactly k peaks which interact with each other and cluster around a local maximum point of V when ε is sufficiently small. The main part of the solutions decays exponentially but the perturbation part decays algebraically at infinity.

We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, where ε is a small positive parameter and a, b > 0 are constants, f ∈ C 1 (ℝ + ,ℝ) is subcritical, V : ℝ 3 → ℝ is a locally Hölder continuous function. We first prove that for ε 0 > 0 sufficiently small, the above problem has a weak solution u ε with exponential decay at infinity. Moreover, u ε concentrates around a local minimum point of V in Λ as ε → 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.

In this paper we prove an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given by where Ω is a smooth bounded domain in ℝ N , N ≥ 3 and 2 ≤ p < N. The function a : ℝ + → ℝ + grows like as t → +∞ for some p ≤ q < N, the case q = p meaning that a is bounded away from zero and infinity. The nonlinearity f : ℝ → ℝ grows like |t| m-1 at infinity with Moreover, we find that u has exactly two nodal domains or changes sign exactly once in Ω. The functions f and a satisfy suitable additional growth and monotonicity conditions which allow this result to extend previous ones to a larger class of p&q type problems. The proof is based on a minimization argument and a variant of quantitative deformation lemma.