Nonlinear Singular Elliptic Equations

Abstract

We improve the Poincaré inequality, the Sobolev embedding theorem and the Trudinger embedding theorem and prove a mountain‐pass theorem. Applying these results we study a nonlinear singular mixed boundary problem.

A MULTIPLICITY RESULT FOR A CLASS OF EQUATIONS OFp-LAPLACIAN TYPE WITH SIGN-CHANGING NONLINEARITIES

“…To overcome this difficulty, we shall use a weak version of the Mountain pass theorem introduced by Duc [3]. Now we introduce the following concept of weakly continuously differentiability due to Duc.…”

“…For the use of Proposition 1, we refer the reader to [3,12,13]. We end this section by studying some certain properties of the functional λ given by (1.5) but we first recall some results which will be used throughout this work.…”

“…In the next part of this paper, we shall show the existence of the second solution u 2 = u 1 by using the Mountain pass theorem introduced in [3]. To this purpose, we first fix λ λ and set Proof.…”

“…We point out the fact that in [4, 13], D. M. Duc and N. T. Vu have studied the following Dirichlet elliptic problem, Mountain pass theorem introduced in [3]. Then, H. Q. Toan and Q.-A.…”

“…Ngô [12] gave some multiplicity results in the case when f (x, u) = h(x)|u| r−1 u + g(x)|u| s−1 u. Using the Mountain pass theorem in [3] combined with Ekeland's variational principle in [5] they proved that problem (1.3)-(1.4) has at least two weak solutions.…”