We consider a class of variational systems in R N of the form where a; b : R N ! R are continuous functions which are coercive; i.e., a(x) and b(x) approach plus innnity as x approaches plus innnity. Under appropriate growth and regularity conditions on the nonlinearities Fu(:) and Fv (:), the (weak) solutions are precisely the critical points of a related functional deened on a Hilbert space o...

We propose a constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem providing an algorithm, based on a bisection method, for its implementation. The efficiency of our algorithm, particularly suitable for problems in high dimensions, consists in the low number of flow lines to be computed for its convergence; for this reason it improves the one currently used and proposed by Y.S....

This paper is devoted to study the multiplicity of nontrivial nonnegative or positive solutions to the following systems    −4pu = λa1(x)|u|q−2u + b(x)Fu(u, v), in Ω, −4pv = λa2(x)|v|q−2v + b(x)Fv(u, v), in Ω, u = v = 0, on ∂Ω, where Ω ⊂ R is a bounded domain with smooth boundary ∂Ω; 1 < q < p < N , p∗ = Np N−p ; 4pw = div(|∇w|p−2∇w) denotes the p-Laplacian operator; λ > 0 is a positive pa...

We discuss Neumann and Robin problems driven by the p-Laplacian with jumping nonlinearities. Using sub-sup solutionmethod, Fucı́k spectrum, mountain pass theorem, degree theorem together with suitable truncation techniques, we show that the Neumann problem has infinitely many nonconstant solutions and the Robin problem has at least four nontrivial solutions. Furthermore, we study oscillating equ...

Let G = (V, E) be a locally finite graph, whose measure μ(x) have positive lower bound, and ∆ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some nonlinear equations, namely ∆u + hu = f (x, u), x ∈ V . In particular, we prove that if h and f satisfy certain assumptions, then the above mentioned equation has stric...

where K, h ∈ L(R), a(x) is a positive bounded function, and f(s) is asymptotically linear with respect to s at infinity, that is f(s)/s goes to a constant as s → +∞. Under suitable assumptions on K, a and f , we prove that the problem (P) has at least two positive solutions for |K|2 and |h|2 small by using Ekeland’s variational principle and Mountain pass theorem.

In this paper we will investigate the existence of multiple solutions for the problem (P ) −∆pu+ g(x, u) = λ1h(x) |u|p−2 u, in Ω, u ∈ H 0 (Ω) where ∆pu = div ( |∇u|p−2 ∇u ) is the p-Laplacian operator, Ω ⊆ IR is a bounded domain with smooth boundary, h and g are bounded functions, N ≥ 1 and 1 < p < ∞. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existe...

We study a class of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving the p(x)-Laplace operator and powertype nonlinear terms with variable exponent. The main results of this articles establish sufficient conditions for the existence of nontrivial weak solutions, in relationship with the values of certain real parameters. The proofs combine the Ekeland variational...

Let Ω be a bounded domain in R . In this paper, we consider the following nonlinear elliptic equation of N -Laplacian type: (0.1) { −∆Nu = f (x, u) u ∈ W 1,2 0 (Ω) \ {0} when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosett...

Existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature. We initiate the corresponding study of higher critical points by considering a superlinear free boundary problem related to plasma confinement. The associated energy functional is nondifferentiable, and therefore standard variational methods cannot be used directly to prove...