Solutions to critical elliptic equations with multi-singular inverse square potentials

Abstract

Let be an open-bounded domain in R N (N 3) with smooth boundary * . We are concerned with the multi-singular critical elliptic problemwhere i ∈ R, 2 * = 2N N−2 , a i ∈ (1 i k) and Q(x) is a positive bounded function on . Using Moser iteration, we prove the asymptotic behavior of solutions for ( * ) at points a i (1 i k). By exploiting the effect of the coefficient of the critical nonlinearity, we, by means of a variational method, establish the existence of positive and sign-changing solutions for problem ( * ).

Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential

“…As in [13] one of the major difficulty to prove the existence of infinitely many solutions for (1.1) by using the variational methods is that I (u) does not satisfy the Palais-Smale condition for large energy level, since 2 * is the critical exponent for the Sobolev embedding from H 1 ( ) to L q ( ). Another difficulty is that, unlike in [13], every nontrivial solution of (1.1) is singular at x = 0 if μ = 0 (see [8,9]). So, different techniques are needed to deal with the case μ > 0.…”

Solutions of the quasilinear elliptic problem with a critical Sobolev–Hardy exponent and a Hardy-type term

“…On the other hand, it should be mentioned that, in recent years people had paid much attention to the singular semilinear problems involving Hardy inequality and Sobolev-Hardy inequality, many results were obtained, which give us very good insight to the singular semilinear problems, see for example [5][6][7][8][9]14,16,17] and references therein. However, compared with the semilinear case, the results for singular quasilinear equations are less, many challenging quasilinear problems involing Hardy inequality and Sobolev-Hardy inequality remain unknown and need to be investigated further.…”

Solutions of a quasilinear elliptic problem involving a critical Sobolev exponent and multiple Hardy-type terms

“…Many important results on the nontrivial solutions of (1.11) were obtained in these publications and these results give us very good insight into the problem. Most recently, the authors of [5] and [9] studied the following semilinear problem with multiple Hardy-type terms:…”

“…Stimulated by [1,5,9,13], in this paper we shall study the existence of solutions of problem (1.1) and give some positive answers to the above question. To our knowledge, there are no results on the existence of nontrivial solutions for (1.1) for k 2.…”