Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball

Abstract

Let B 1 be the unit open ball with center at the origin in R N , N 2. We consider the following quasilinear problem depending on a real parameter λ > 0:where f (t) is a nonlinearity that grows like e t N/N −1 as t → ∞ and behaves like t α , for some α ∈ (−∞, 0), as t → 0 + . More precisely, we require f to satisfy assumptions (A1) and (A2) listed in Section 1. For such a general nonlinearity we show that if λ > 0 is small enough, (P λ ) admits at least one weak solution (in the sense of distributions). We further study the question of uniqueness and multiplicity of solutions to (P λ ) when Ω = B 1 under additional structural conditions on f (see assumptions (A3)-(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)-(A5), we prove uniqueness of solution to (P λ ) for all λ > 0 small whereas under (A6), (A7) or (A8), we show multiplicity of solutions to (P λ ) for all λ > 0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition lim inf t→∞ h(t)te εt 1/(N −1) = ∞ ∀ε > 0.

Radial singular solutions for the N-Laplace equation with exponential nonlinearities

“…It can be checked thatf 0 also satisfies assumptions (A1)-(A5) in [11]. Therefore, again by [11,Proposition 4.2], we get S m < ∞. Since y * is nondecreasing, we obtain that y * (t) → ∞ as t → ∞.…”

Existence of positive solutions for a class of singular and quasilinear elliptic problems with critical exponential growth

“…In [20] Giacomoni, Prashanth and Sreenadh studied a problem with N-Laplacian such that the nonlinearity grows like exp(|t| N/N −1 ) at infinity and like 1 t α at the origin. A similar problem with the Laplacian operator in R 2 was studied by Saoudi and Kratou in [35].…”

“…But the nonlinearities have polynomial growth. 2) In [16], [20], [35] and [36] were studied the singular case with a nonlinearities with exponential growth. However, here we study problems with a general operator which brings some technical difficulties.…”

Kirchhoff type equations with strong singularities

“…(see, e.g. [1][2][3][4][5][6][7][8][9][10][11][12][14][15][16][17][18][19][20][21][22][23][24][25][26][28][29][30][31][32][33][34][36][37][38][39][40][41][42] and references therein). For the singular boundary theory, we refer the reader to the books by Agarwal and O'Regan [1], and Hernández and Mancebo [25] for an excellent introduction to the subject.…”

“…As is well known, for any Ω ⊂⊂ Ω, where Ω is a bounded open set in R N with N ≥ 1, we can construct some ψ(x) ∈ C ∞ 0 (Ω) be such that 0 ≤ ψ(x) ≤ 1 on Ω, ψ(x) ≡ 1 on Ω and |D β ψ(x)| ≤ C(N,|β|)[dist(Ω ,∂Ω)] |β| on Ω by taking advantage of mollifiers. Thus the above equation leads, thanks to(16), to the definition of ψ δ (x) and to the facts that g(x) ≥ 0, −γ < 0, toΩ |x| −α u −γ + g(x)u q ψ δ dx = B 2δ (0)\B δ (0) |x| −α u −γ + g(x)u q ψ δ dx ≥ B 2δ (0)\B δ (0) |x| −α u −γ ψ δ dx ≥ sup Ω u −γ B 2δ (0)\B δ (0) |x| −α ψ δ dx δ dx = (a + b||u|| 2 ) Ω u · −∆ψ δ dx ≤ (a + b||u|| 2 ) Ω u · |∆ψ δ |dx = (a + b||u|| 2 ) B 2δ (0)\B δ (0) u · |∆ψ δ |dx ≤ sup Ω u (a + b||u|| 2 ) C(N ) δ 2 2 N − 1 ω N δ N ,…”