Existence and multiplicity of solutions to a singular elliptic system with critical Sobolev–Hardy exponents and concave–convex nonlinearities

Abstract

a b s t r a c tThis paper is concerned with a singular elliptic system, which involves the CaffarelliKohn-Nirenberg inequality and critical Sobolev-Hardy exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.

Existence of solutions of fractional p‐Laplacian systems with different critical Sobolev‐Hardy exponents

“…The proofs of Theorem 1 are obtained by applying variational arguments inspired by other studies. [22][23][24]33 Theorem 2. Suppose N > sp, s∈(0,1), and 0 < < ∞, then ( , , 1 , 2 , , ) has the minimizers (U (x), min U (x)), ∀ >0, where U (x) are defined as in (15).…”

“…The proofs of Theorem 1 are obtained by applying variational arguments inspired by other studies 22‐24,33 …”

“…To continue, by the same as that in Nyamoradi, 33 Let be a minimizing sequence for S ∗ . Let e 1 , e 2 >0 be chosen later.…”

Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents inRN

“…Via the analytic techniques of Nehari manifold and variational methods, the author proved that the system (4) admits at least two nontrivial nonnegative solutions if the pair of parameters ( , ) belongs to a certain subset of R 2 . Very recently, Nyamoradi [16], Lü and Xiao [17], and Li and Gao [18] generalized the corresponding results of [15] to the nonlinear singular elliptic systems involving critical Hardy-Sobolev exponents. Other results about existence and multiplicity of nontrivial solutions, also for related elliptic systems, can be seen in [19][20][21][22][23] and the references therein.…”

Symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents

“…Applying the analytic techniques of Nehari manifold, the author proved that the system (1.4) had at least two nonnegative solutions if the parameters λ and σ satisfied an appropriate condition. Subsequently, Nyamoradi [15] generalized some results of [14] to the singular elliptic systems involving critical Hardy-Sobolev exponents. Very recently, Huang and Kang [16] studied the existence and asymptotic properties of positive solutions of the following critical singular elliptic systems 2), and α, β > 1 satisfy α + β = 2 * .…”