By a variant version of mountain pass theorem, the existence and multiplicity of solutions are obtained for a class of superlinear p-Laplacian equations: −Δ p u = f (x,u). In this paper, we suppose neither f satisfies the superquadratic condition in Ambrosetti-Rabinowitz sense nor f (x,t)/|t| p−1 is nondecreasing with respect to |t|. This is an open access article distributed under the Creative...

Since A. Ambrosetti and P.H. Rabinowitz proposed the mountain pass theorem in 1973 (see [1]), critical point theory has become one of the main tools for finding solutions to elliptic problems of variational type. Especially, elliptic problem (1.2) has been intensively studied for many years. One of the very important hypotheses usually imposed on the nonlinearities is the following Ambrosetti-R...

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.