We consider the following nonlinear Schrödinger equation { ∆u− (1 + δV )u + f(u) = 0 in R , u > 0 in R , u ∈ H(R ) where V is a continuous potential and f(u) is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a δ0 such that for 0 < δ < δ0, the above problem has infinitely many positive solu...

This paper deals with a vector polynomial optimization problem over basic closed semi-algebraic set. By invoking some powerful tools from real geometry, we first introduce the concept called tangency varieties; obtain relationships of Palais–Smale condition, Cerami M-tameness, and properness related to considered problem, in which condition Mangasarian–Fromovitz constraint qualification at infi...

Let X be a real Banach space and Φ ∈ C 1 (X, R) a function with a mountain pass geometry. This ensures the existence of a Palais-Smale, and even a Cerami, sequence {u n } of approximate critical points for the mountain pass level. We obtain information about the location of such a sequence by estimating the distance of u n from S for certain types of set S as n → ∞. Under our hypotheses we can ...

LRES, Faculty of Sciences Tunis, University Tunis El Manar, Tunisia EM [email protected] KW Asymptotically linear % variational method Schr?dinger equations Cerami sequence KR nema In this paper, we investigate a quasilinear problem under Dirichlet boundary condition in regular domain with asymptotically nonlinearities. We use version the mountain pass theorem to prove existence solution ...

We are concerned with the study of existence and multiplicity solutions for Dirichlet boundary value problems, involving $ ( p( m ), \, q( ) )- equation nonlinearity is superlinear but does not fulfil Ambrossetti-Rabinowitz condition in framework Sobolev spaces variable exponents a complete manifold. The main results proved using mountain pass theorem Fountain Cerami sequences. Moreover, an exa...

In this paper we study the multiplicity of solutions for a class of eigenvalue problems for hemivariational inequalities in strip-like domains. The first result is based on a recent abstract theorem of Marano and Motreanu, obtaining at least three distinct, axially symmetric solutions for certain eigenvalues. In the second result, a version of the fountain theorem of Bartsch which involves the ...

Abstract The aim of this paper is to examine the existence at least two distinct nontrivial solutions a Schrödinger-type problem involving nonlocal fractional $p(\cdot )$ p ( ⋅ ) -Laplacian with concave–convex nonlinearities when, in general, nonlinear term d...

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...

A. We consider the semi-linear elliptic PDEs driven by the fractional Laplacian: { (−∆)su = f (x, u), in Ω, u = 0, in Rn\Ω. By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais-Smale condition without AmbrosettiRabinowitz condition for non-local el...

Real innovations in medicine and science are historic and singular; the stories behind each occurrence are precious. At Molecular Medicine we have established the Anthony Cerami Award in Translational Medicine to document and preserve these histories. The monographs recount the seminal events as told in the voice of the original investigators who provided the crucial early insight. These essays...