Infinitely many solutions for an elliptic Neumann problem involving critical Sobolev growth

Abstract

In this paper, we prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth:where Ω is a bounded domain in R N with C 3 boundary, N 3, ν is the outward unit normal of ∂Ω, 2 * = 2N N−2 , and g(t) = μ|t| p−2 t − t, or g(t) = μt, where p ∈ (2, 2 * ), μ > 0 are constants.We obtain the existence of infinitely many solutions under certain assumptions on N, p and ∂Ω. In particular, if g(t) = μt with μ > 0, N 7, and Ω is a strictly convex domain, then the problem has infinitely many solutions.

Infinitely many solutions to elliptic systems with critical exponents and Hardy potentials

“…Devillanova and Solimini in are the first to obtain infinitely many solutions to semilinear problem with Sobolev critical growth A major step in is to prove that solutions for the approximating system with subcritical growth are bounded in and uniformly bounded in L ∞ (Ω). Then, using the idea in , Cao and Yan obtained infinitely many solutions for semilinear elliptic equations with Hardy term in and for Neumann boundary problem in .…”

“…/. Then, using the idea in [7], Cao and Yan obtained infinitely many solutions for semilinear elliptic equations with Hardy term in [8] and for Neumann boundary problem in [9].…”

Existence and multiplicity of positive solutions for a class of elliptic equations involving critical Sobolev exponents

“…Later, Ambrosetti et al [1] have considered problem (1.3) with 0 < q < 1, and obtained two positive solutions by the sub-supersolution method and the variant mountain pass theorem. After that, many papers have studied the existence and multiplicity of positive solutions of the problem with critical exponent, such as [2,3,[5][6][7][8][9][10][11][12][13][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][31][32][33][35][36][37]. Particularly, [9,17,21,23] have considered problem (1.1) with 0 ≤ q < 1.…”

Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

“…Devillanova and Solimini are the first to obtain infinitely many solutions to semilinear problem with Sobolev critical growth A major step in is to prove that solutions for the approximating system with subcritical growth are bounded in and uniformly bounded in L ∞ (Ω). Then, using the idea in , Cao and Yan obtained infinitely many solutions for semilinear elliptic equations with the Hardy term in and for the Neumann boundary problem in .…”

“…/. Then, using the idea in [7], Cao and Yan obtained infinitely many solutions for semilinear elliptic equations with the Hardy term in [8] and for the Neumann boundary problem in [9].…”