Existence and Non-Existence of Radial Solutions for Elliptic Equations with Critical Exponent in E2
نویسنده
چکیده
where 2* = 2 N / ( N 2) is the critical Sobolev exponent for the embedding H’(52) C L2* (0). Criticality and subcriticality play important rBles concerning the solvability of equation (1.1). Using (by now classical) variational methods one sees that equation (1.1) is solvable i f f has subcritical growth (and satisfies some additional hypotheses), while for f(u) = IuIP-~u the well-known Pohoiaev identity shows that the solvability of equation (1.1) is lost if p reaches 2* (and R is starshaped). In their important paper, [6], Brezis-Nirenberg showed that for N E 4 solvability is regained if one adds a lower order (linear) term to the critical term. More precisely they showed that for f ( r ) = Ar + lr12*-2r one has ( A , denotes the 1st eigenvalue of -A on HA(R)):
منابع مشابه
The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent
In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.
متن کاملExistence Results for a Dirichlet Quasilinear Elliptic Problem
In this paper, existence results of positive classical solutions for a class of second-order differential equations with the nonlinearity dependent on the derivative are established. The approach is based on variational methods.
متن کاملExistence of three solutions for a class of quasilinear elliptic systems involving the $p(x)$-Laplace operator
The aim of this paper is to obtain three weak solutions for the Dirichlet quasilinear elliptic systems on a bonded domain. Our technical approach is based on the general three critical points theorem obtained by Ricceri.
متن کاملExistence of at least three weak solutions for a quasilinear elliptic system
In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...
متن کاملExistence of ground state solutions for a class of nonlinear elliptic equations with fast increasing weight
This paper is devoted to get a ground state solution for a class of nonlinear elliptic equations with fast increasing weight. We apply the variational methods to prove the existence of ground state solution.
متن کاملQuasilinear Schrödinger equations involving critical exponents in $mathbb{textbf{R}}^2$
We study the existence of soliton solutions for a class of quasilinear elliptic equation in $mathbb{textbf{R}}^2$ with critical exponential growth. This model has been proposed in the self-channeling of a high-power ultra short laser in matter.
متن کامل