On the singular second-order elliptic equation

Journal of Physics: Conference Series, 2014

Given a n−dimensional compact Riemannian manifold (M, g) with n ≥ 5, we consider the following semi-linear elliptic equation : Pg(u) := ∆ 2 g u + divg (a(x)∇gu) + b(x)u = f (x) |u| N −2 u + λh(x) |u| q−2 u where the functions a, b and h are in suitable Lebesgue spaces, 2 < q < N and λ > 0 a real parameter, f is a smooth positive function and the operator Pg is coercive. Under some additional conditions, we obtain results concering the existence of strong solutions of the above equation in H 2 2 (M).

Electronic Journal of Differential Equations, 2007

Electronic Journal of Differential Equations, Vol. 2007(2007), No. 84, pp. 1–6. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ... AN EXISTENCE RESULT FOR ELLIPTIC PROBLEMS ...

Journal of Mathematical Analysis and Applications

We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary condition. We prove uniform Lebesgue estimates and we determine the critical dimensions for these problems with nonlinearities of the type Gelfand, MEMS and power case. As an application, we show that extremal solutions are classical whenever the dimension of the manifold is below the critical dimension of the associated problem. Moreover, we analyze the branch of minimal solutions and we prove multiplicity results when the parameter is close to critical threshold and we obtain uniqueness on it. Furthermore, for the case of Riemannian models we study properties of radial symmetry and monotonicity for semistable solutions.

Journal of Differential Equations, 2010

In this paper we deal with a nonlinear elliptic problem, whose model is

Complex Variables and Elliptic Equations, 2014

Using the method of Nehari manifold, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singulatities and with critical Sobolev growth.

Rendiconti Lincei - Matematica e Applicazioni, 2000

We deal with the semi-linear elliptic problem

Differential and Integral Equations, 2008

Proceedings of the Steklov Institute of Mathematics, 2006

In this paper we consider nonlinear elliptic equations of the form −∆u = g(u) , in Ω, u = 0 , on ∂Ω , and Hamiltonian type systems of the form :        −∆u = g(v) , in Ω, −∆v = f (u) , in Ω, u = 0 and v = 0 , on ∂Ω, where Ω is a bounded domain in R 2 , and f , g ∈ C(R) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f , g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev-Trudinger type inequalities. We discuss existence and nonexistence results related to critical growth for the equation and the system. The natural framework for such equations and systems are Sobolev spaces, which give in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results.

Nonlinear Analysis: Theory, Methods & Applications, 1981

We give a classification of isolated singularities of any solution of the equation-Au + u julq-' = 0. If 0 is the singular point, we prove that there exist two types of singularities when 1 < 4 < N/(N-2) and that we have essentially the following in 0 (i) either u(x) h k [q,NI~j-2'(q-1), with Iq, N = ((2/(q-1))(2q/(q-1)-N))"(y-l', (ii) or u(x) N c&4, where c is any constant and cp the fundamental solution of the Laplacian in W'.

Journal of Mathematical Analysis and Applications, 2011

In this paper we apply minimax methods to obtain existence and multiplicity of weak solutions for singular and nonhomogeneous elliptic equation of the form − N u = f (x, u) |x| a + h(x) in Ω, where u ∈ W 1,N 0 (Ω), N u = div(|∇u| N−2 ∇u) is the N-Laplacian, a ∈ [0, N), Ω is a smooth bounded domain in R N (N 2) containing the origin and h ∈ (W 1,N 0 (Ω)) * = W −1,N is a small perturbation, h ≡ 0. Motivated by a singular Trudinger-Moser inequality, we study the case when f (x, s) has the maximal growth on s which allows to treat this problem variationally in the Sobolev space W 1,N 0 (Ω).