Concentration Phenomena for the Paneitz Curvature Equation in ℝ^N

Abstract

In this paper, we study the Paneitz curvature problemΔwhere

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

“…Recently, the nonlinear biharmonic equation has been extensively studied. We refer readers to existing studies() for the subcritical case and other studies() for the critical case. Now let us briefly comment some known results of them.…”

“…Moreover, they proved that there is no solution, which concentrates at one point. When V ( x ) = 1 + τ K ( x ), ϵ = 0, Liu proved the existence of two peak solutions for the above problem, provided τ is small enough under some appropriate conditions on K . For the supercritical case, to our knowledge, there are no results about the high‐energy solutions for the biharmonic Schrödinger equation involving supercritical Sobolev exponent.…”

Infinitely many peak solutions for a biharmonic equation involving critical exponent

“…Ben Ayed, El Mehdi and Hammami [16] obtained some existences of single-peak solutions by the theory of critical points at infinity under some assumptions on K.y/. However, much less is known about the multiplicity of the solutions for problem (1.1); for other results related to critical biharmonic problems, see for example, [17][18][19][20][21][22][23][24][25][26][27].…”

Concentration of solutions for the fractional Nirenberg problem

“…The proof of our results is inspired by the methods of [10,32,35]. More precisely, we will use a reduction argument similar to [32,35] to prove Theorem 1.1, see also [7,8,9,29,30,34]. It is worthwhile to point out that the solutions constructed in this paper have exactly two maximum points, the distance between which is very large.…”