A deformation lemma for functionals which are the sum of a locally Lipschitz continuous function and of a concave, proper and upper semicontinuous function is established. Some critical point theorems are then deduced and an application to a class of elliptic variational-hemivariational inequalities is presented. Introduction It is by now well known that the Mountain Pass Theorem of Ambrosetti ...

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...

We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang’s version of the mountain pass lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth f...

We present a version of the classical Mountain Pass Lemma and explain how to combine it with constraint qualifications to prove that nonlinear programming problems have a unique local minimizer.

In this paper, by using concentration-compactness principle and a new version of the symmetric mountain-pass lemma due to Kajikiya (J Funct Anal 225:352–370, 2005), infinitely many small solutions are obtained for a class of quasilinear elliptic equation with singular potential −∆pu− μ |u| p−2u |x|p = |u|p(s)−2u |x|s + λf(x, u), u ∈ H 1,p 0 (Ω). Mathematics Subject Classification (2000). 35J60,...

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...

In this paper we study a quasilinear elliptic system coupled by Schrödinger equation with p-Laplacian operator and Poisson equation. Some scaling transformation ingenious methods are applied to produce the bounded Palais-Smale sequences existence of nontrivial solutions for is obtained mountain pass theorem.

We study the existence of non-trivial solutions for a class of problems containing in particular the following system of two coupled semilinear Poisson equations: (I) 8 > < > : ? v = f(u) in ; ? u = g(v) in ; u = v = 0 on @: (1) (2) (3) Here is a bounded domain in R N with a smooth boundary, and is the Laplace operator. Problem (I) allows for a variational formulation, i.e. solutions arise as c...