1,721,034 research outputs found
The Yamabe equation in a non-local setting
No full textAim of this paper is to study an elliptic equation driven by a general non-local integrodifferential operator and depending on a real parameter, in an open bounded set with Lipschitz boundary. In this framework, in the existence result proved along the paper, we show that our problem admits a non-trivial solution for any positive parameter provided it is different from the eigenvalues of the non-local operator. This result may be read as the non-local fractional counterpart of the one obtained by Capozzi, Fortunato and Palmieri and by Gazzola and Ruf for the classical Laplace equation with critical nonlinearities. In this sense the present work may be seen as the extension of some classical results for the Laplacian to the case of non-local fractional operators
Mountain Pass and Linking methods for elliptical semilinear variational inequalities: results of existence, stability and multiplicity
No full textIn this thesis we will deal with semilinear elliptic variational inequalities and we will discuss some results about existence,
stability, and multiplicity obtained via variational methods. The main purpose of this thesis is to consider a completely new approach with respect to the papers already appeared in the literature. The fundamental tools used in this thesis to look for nontrivial
solutions for the problem under consideration are a penalization method and some minimax theorems
Existence results for semilinear elliptic variational inequalities with changing sign nonlinearities
No full textIn this paper we present some existence results for a class of semilinear elliptic variational inequalities, depending on a real parameter lambda, with changing sign nonlinearities. The fundamental tool to prove the existence result is a penalization method combined with the Mountain Pass Theorem and the Linking Theorem, respectively in the case lambda = lambda (1), where lambda(1) is the first eigenvalue of the uniformly elliptic operator A involved in the variational inequality
A critical fractional Laplace equation in the resonant case
No full textIn this paper we complete the study of the a non-local fractional equation involving critical nonlinearities depending on a real parameter, started in some recent papers by the same authors (joint with other coauthors). Aim of this paper is to study this critical problem in the special case when and the parameter is an eigenvalue of the fractional Laplace opertaor with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a nontrivial solution, so that with the results obtained in some recent papers, we are able to show that this critical problem admits a nontrivial solutionunder suitable assumptions on the dimension of the space and on the parameter appearing in the equation. In this way we extend completely the famous result of Brezis and Nirenberg for the critical Laplace equation to the non-local setting of the fractional Laplace equation
Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity
No full textIn this paper we discuss the existence of infinitely many solutions for a
nonlocal, nonlinear equation with homogeneous Dirichlet boundary data. Adapting the classical variational techniques used in
order to study the standard Laplace equation with subcritical growth nonlinearities to the nonlocal framework, along the present paper we prove that this problem admits infinitely many weak solutions uk, with the property that their Sobolev norm goes to infinity, provided the exponent satisfies suitable assumptions. In this sense, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators
A semilinear elliptic PDE not in divergence form via variational methods
No full textIn this paper we consider a semilinear equation driven by an operator not in divergence form. Precisely, the principal part of the operator is in divergence form, but it has also a lower order term depending on Du. While the right-hand side of the equation satisfies superlinear and subcritical growth conditions at zero and at infinity. The problem has not a variational structure, but, despite that, we use variational techniques in order to prove an existence and regularity result for the equation
NONEXISTENCE FOR P-LAPLACE EQUATIONS WITH SINGULAR WEIGHTS
No full textAim of this paper is to give some nonexistence results of nontrivial solutions for a quasilinear elliptic equations with singular weights in R^n / {0}. The main tool for deriving nonexistence theorems for the equations is a Pohoaev-type identity. We first show that such identity holds true for weak solutions sufficiently smooth. Then, under a suitable growth condition on the nonlinearity, we prove that every weak solution has thhe required regularity, so that the Pohosaev-type identity can be applied. From this identity we derive some nonexistence results, improving several theorems already appeared in the literature. In particular, we discuss the case when h and f are pure powers
The Brezis-Nirenberg result for the fractional Laplacian
Full text linkThe aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). In this paper we first study the problem in a general framework; indeed we consider an equation driven by a general non-local integrodifferential operator and in presence of a lower order perturbation of the critical power. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for an equation driven by the fractional Laplacian (-Delta)(s); that is, we show that if lambda(1,s) is the first eigenvalue of the non-local operator (-Delta)(s) with homogeneous Dirichlet boundary datum, then for any lambda is an element of (0, lambda(1,s)) there exists a non-trivial solution of the above model equation, provided n >= 4s. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators
On the existence of solutions for nonlinear impulsive periodic viable problems
No full textIn this paper we prove the existence of periodic solutions for nonlinear impulsive viable problems monitored by differential inclusions of the type x' (t) is an element of F (t, x(t)) + G (t, x(t)). Our existence theorems extend, in a broad sense, some propositions proved in [10] and improve a result due to Hristova-Bainov in [13]
Soliton dynamics for a general class of Schrodinger equations
No full textThe soliton dynamics for a general class of nonlinear focusing Schrodinger problems in presence of non-constant external (local and nonlocal) potentials is studied by taking as initial datum the ground state solution of ail associated autonomous elliptic equation. (C) 2009 Elsevier Inc. All rights reserved
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