Multi‐peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents

Abstract

We study the following Kirchhoff equation: −aε2+bε4−N∫RN|∇u|2△u=g(x,u),a,b>0,N≥3,(K) where gfalse(x,ufalse)=Kfalse(xfalse)false|ufalse|2∗−2u0.3emor0.3emgfalse(x,ufalse)=−Vfalse(xfalse)u+false|ufalse|p−2ufalse(2<p<2∗false) is critical or subcritical growth. By introducing suitable scalings, false(scriptKfalse) is transformed into a semilinear elliptic equation coupled with an algebraic equation of the scaling parameter. Upon which, we prove the existence of multi‐peak solutions of false(scriptKfalse) concentrating around the critical points of Kfalse(xfalse) or Vfalse(xfalse). Moreover, we prove that false(scriptKfalse) has only trivial solution if N≥4 and b>0 is suitably large.

Positive normalized solution to the Kirchhoff equation with general nonlinearities

“…The asymptotic behavior of the solutions is also a research interest of scholars. In [12,16,48,49], the authors considered (1.1) with very general nonlinearities, even involving the Sobolev critical exponent, and studied the existence, multiplicity and concentration behavior of positive solutions. For the bounded states, we refer to [17,18].…”

Positive normalized solution to the Kirchhoff equation with general nonlinearities of mass super-critical

“…The asymptotic behavior of the solutions is also a research interest of scholars. In [2,4,19,26], the authors considered (1.1) with very general nonlinearities, even involving the Sobolev critical exponent, and studied the existence, multiplicity and concentration behavior of positive solutions. For the bounded states, we refer to [9,11].…”

Existence and uniqueness of minimizers for L2‐constrained problems related to fractional Kirchhoff equation

“…3 Several works considered the existence of solutions for equation (12) with s = 1, see other studies for the subcritical case [4][5][6][7][8][9][10] and for the critical case. [11][12][13] The asymptotic properties for minimizers of (1.1) with s = 1 can see Guo et al 14 When 0 < s < 1 and a = 1, b = 0, equation (12) reduced to the fractional Laplacian equation.…”