In this article we consider the differential inclusion − div(|∇u|p(x)−2∇u) ∈ ∂F (x, u) in Ω, u = 0 on ∂Ω which involves the p(x)-Laplacian. By applying the nonsmooth Mountain Pass Theorem, we obtain at least one nontrivial solution; and by applying the symmetric Mountain Pass Theorem, we obtain k-pairs of nontrivial solutions in W 1,p(x) 0 (Ω).

By using the Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order nonlinear difference equation ∆ [p(t)∆u(t − 1)] + f(t, u(t)) = 0 has at least one homoclinic orbit, where t ∈ Z, u ∈ R.

Abstract. We consider a class of nonlinear Dirichlet problems involving the p(x)–Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The proof relies on the Mountain Pass Theorem.

We study the nonuniformly elliptic, nonlinear system − div(h1(x)∇u) + a(x)u = f(x, u, v) in R , − div(h2(x)∇v) + b(x)v = g(x, u, v) in R . Under growth and regularity conditions on the nonlinearities f and g, we obtain weak solutions in a subspace of the Sobolev space H1(RN , R2) by applying a variant of the Mountain Pass Theorem.

We establish existence and multiplicity theorems for a Dirichlet boundary-value problem at resonance. This problem is a nonlinear subcritical perturbation of a linear eigenvalue problem studied by Cuesta, and includes a sign-changing potential. We obtain solutions using the Mountain Pass lemma and the Saddle Point theorem. Our paper extends some recent results of Gonçalves, Miyagaki, and Ma.

We establish the existence of two nontrivial solutions to semilinear elliptic systems with superquadratic and subcritical growth rates. For a small positive parameter λ, we consider the system −∆v = λf(u) in Ω, −∆u = g(v) in Ω, u = v = 0 on ∂Ω, where Ω is a smooth bounded domain in R with N ≥ 1. One solution is obtained applying Ambrosetti and Rabinowitz’s classical Mountain Pass Theorem, and t...

In this paper we study the existence of periodic solutions of the fourth-order equations u − pu′′ − a x u + b x u3 = 0 and u − pu′′ + a x u − b x u3 = 0, where p is a positive constant, and a x and b x are continuous positive 2Lperiodic functions. The boundary value problems P1 and P2 for these equations are considered respectively with the boundary conditions u 0 = u L = u′′ 0 = u′′ L = 0. Exi...

These notes are designed to serve as a template of a LaTeX article. In the process we will describe some notions of Geometric Analysis pertaining to the Mountain Pass Theorem. Little attempt was made to be a publishable set of notes, but instead to provide examples of commonly used commands, environments, and symbols in LaTeX.

An integral inequality is deduced from the negation of the geometrical condition in the bounded mountain pass theorem of Schechter, in a situation where this theorem does not apply. Also two localization results of non-zero solutions to a superlinear boundary value problem are established.

The existence of homoclinic orbits is obtained for a class of the second order Hamiltonian systems ü(t)−L(t)u(t)+∇W (t,u(t)) = 0, ∀t ∈ R , by the mountain pass theorem, where W(t,x) needs not to satisfy the global (AR) condition. Mathematics subject classification (2000): 34C37, 37J45, 47J30, 58E05.