In this study, the authors utilize mountain pass lemma, variational methods, regularization technique, and Lyapunov function method to derive unique existence of positive classical stationary solution a single-species ecosystem. Particularly, geometric characteristic saddle point in lemma guarantees that equilibrium is ground state Based on obtained uniqueness result, use globally exponential s...

During the last twenty years, many minimax theorems that have proved to be very useful tools in finding critical points of functionals have been established. They have all in common a geometric intersection property known as the linking principle. Our purpose in this paper is to give a linking theorem that strengthens and unifies some of these works. We think essentially to Ambrosetti-Rabinowit...

The definition of the weak slope of continuous functions introduced by Degiovanni and Marzocchi (cf. [8]) and its interrelation with the notion “steepness” of locally Lipschitz functions are discussed. A deformation lemma and a mountain pass theorem for usco mappings are proved. The relation between these results and the respective ones for lower semicontinuous functions (cf. [7]) is considered...

We study the existence of multiple nontrivial solutions for nonlinear fourth order discrete boundary value problems. first establish criteria at least two problems and obtain conditions to guarantee that are sign-changing. Under some appropriate assumptions, we further prove have three solutions, which positive, negative, sign-changing, respectively. include examples illustrate applicability ou...

In this paper using the critical point theory of Chang [4] for locally Lipschitz functionals we prove an existence theorem for noncoercive Neumann problems with discontinuous nonlinearities. We use the mountain-pass theorem to obtain a nontrivial solution.

This paper aims to show the existence of nontrivial solutions for discrete elliptic boundary value problems by using the “Mountain Pass Theorem”. Some conditions are obtained for discrete elliptic boundary value problems to have at least two nontrivial solutions. The results obtained improve the consequences of the known literature [Guang Zhang, Existence of nontrivial solutions for discrete el...

In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the p(x)−Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration–compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain...

By means of a three critical points theorem proposed by Brezis and Nirenberg and a general version of Mountain Pass Theorem, we obtain some multiplicity results for periodic solutions of a fourth-order discrete Hamiltonian system ∆u(t− 2) +∇F (t, u(t)) = 0, for all t ∈ Z.

We consider Hardy-Sobolev nonlinear equations on domains with singularities. introduced this problem in Cheikh-Ali [4]. Under a local geometric hypothesis, namely that the generalized mean curvature is negative (see (7) below), we proved existence of extremals for relevant inequality large dimensions. In present paper, tackle question small dimensions was left open. introduce “mass”, global qua...

This work is devoted to study the existence of solutions to equations of the p-Laplacian type in unbounded domains. We prove the existence of at least one solution, and under further assumptions, the existence of in nitely many solutions. We apply the mountain pass theorem in weighted Sobolev spaces.