We consider an eigenvalue problem for a certain type of quasi-linear second-order di®erential equation on the interval ð0;1Þ. Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in H 1 0 ð0;1Þ for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided trave...

The Ambrosetti-Rabinowitz (AR) condition is crucial in variational methods. In this paper we consider a class of p-Laplacian equations without the AR condition. Using Mountain pass lemma and Ekeland variational principle, we obtain the existence and multiplicity of the solutions. These results complement some known results.

The main purpose of this paper is to study the existence of periodic solutions of second order impulsive differential equations with superlinear nonlinear terms. Our result generalizes one of Paul H. Rabinowitz’s existence results of periodic solutions of second order ordinary differential equations to impulsive cases. Mountain Pass Lemma is applied in order to prove our main results. AMS Subje...

where Ω ⊂ R is a bounded domain with Lipschitz boundary and f : R → R is of class C with f(0) = 0. Thus u0 ≡ 0 is a trivial solution of (D) and we are interested in finding and studying nontrivial solutions. One way of obtaining these is to compare the behavior of f near the origin and near infinity. We shall always assume that f grows subcritically at infinity so that variational methods can b...

We consider the motion of n particles under the action of a potential F . Imposing appropriate conditions on F we obtain quasiperiodic solutions using variational methods. A Diophantine condition on the frequency similar to those encountered in KAM theory allows us to establish the necessary properties of the corresponding functional. The solutions are then obtained by means of the mountain pas...

We study the following p-Laplacian equation with nonlinear boundary conditions: -Δ(p)u + μ(x)|u|(p-2)u = f(x,u) + g(x,u),x ∈ Ω, | ∇u|(p-2)∂u/∂n = η|u|(p-2)u and x ∈ ∂Ω, where Ω is a bounded domain in ℝ(N) with smooth boundary ∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f, g do not need to satisfy the (P.S) or (P.S...

The present paper deals with a Kirchhoff problem under homogeneous Dirichlet boundary conditions, set in a bounded smooth domain Ω of R^{N}. The problem studied is a stationary version of the orig inal Kirchhoff equation, involving the p(x)-Lap lacian operator, in the framework of the variable exponent Lebesgue and Sobolev spaces. The question of the existence of weak solutions is treated. Appl...

This article concerns with the problem ?div(jruj m?2 ru) = hu q + u m ?1 ; in R N : Using the Ekeland Variational Principle and the Mountain Pass Theorem, we show the existence of > 0 such that there are at least two non-negative solutions for each in (0;).

Using the mountain pass theorem combined with the Ekeland variational principle, we obtain at least two distinct, non-trivial weak solutions for a class of p(x)-Kirchhoff type equations with combined nonlinearities. We also show that the similar results can be obtained in the case when the domain has cylindrical symmetry.

We apply a recently obtained three-critical-point theorem of B. Ricceri to prove the existence of at least three solutions of certain two-parameter Dirichlet problems defined on the Sierpinski gasket. We also show the existence of at least three nonzero solutions of certain perturbed two-parameter Dirichlet problems on the Sierpinski gasket, using both the mountain pass theorem of Ambrosetti an...