Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros

Iturriaga, Leonelo; Lorca, Sebastián; Massa, Eugenio

doi:10.1016/j.anihpc.2009.11.003

Cited by 15 publications

(32 citation statements)

References 21 publications

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“…We also studied the asymptotical behavior of the solutions as λ tends to zero and to infinity. Later, in , we extended these results to supercritical nonlinearities provided Ω is convex and h does not depend on

x \in Ω

.…”

Section: Introductionmentioning

confidence: 82%

“…We obtained the existence of at least one solution for small positive λ and at least two solutions for large values of λ, assuming a superlinear and subcritical growth at infinity and a p-linear behavior at the origin for the nonlinearity h. We also studied the asymptotical behavior of the solutions as λ tends to zero and to infinity. Later, in [8], we extended these results to supercritical nonlinearities provided is convex and h does not depend on x ∈ .…”

Section: Introductionmentioning

confidence: 83%

“…More precisely, the hypothesis ( H 3) says that the nonlinearity is locally superlinear at

+ \infty

, that is, it is required to be superlinear only in a small interval (which means in a small annulus when considering the radial case). This assumption constitutes one of the main differences when compared to the results in and we cited previously: it is not necessary here to assume superlinearity in the whole of Ω.…”

Section: Introductionmentioning

confidence: 93%

“…An example of a function f which satisfies our broader hypotheses, but not those in and , could be

f (t, u) = \{\begin{matrix} right u {(1 - u)}^{2} & right for u \leq 1, \\ right (e^{u - 1} - u) ϕ (t) & right for u > 1, \end{matrix}

where

ϕ \in C^{0} ([0, 1])

is nonnegative, may be null in some set, but is positive in

[α, β]

.…”

Section: Introductionmentioning

confidence: 99%

“…We observe that other more technical hypotheses were needed in and and are not imposed here, in particular, it is not necessary to assume subcritical growth (as in ) nor the convexity of the domain and that the nonlinear term is independent of

x \in Ω

(as in ).…”

Section: Introductionmentioning

confidence: 99%

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Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros

Iturriaga

Massa

Sánchez

et al. 2014

Mathematische Nachrichten

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We study existence, multiplicity, and the behavior, with respect to λ, of positive radially symmetric solutions of −Δu=λh(x,u) in annular domains in double-struckRN,N≥2. The nonlinear term has a superlinear local growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive and concave function a. For this, we combine several methods such as the sub and supersolutions method, a priori estimates and degree theory.

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x \in Ω

.…”

Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 83%

“…More precisely, the hypothesis ( H 3) says that the nonlinearity is locally superlinear at

+ \infty

Section: Introductionmentioning

confidence: 93%

“…An example of a function f which satisfies our broader hypotheses, but not those in and , could be

f (t, u) = \{\begin{matrix} right u {(1 - u)}^{2} & right for u \leq 1, \\ right (e^{u - 1} - u) ϕ (t) & right for u > 1, \end{matrix}

where

ϕ \in C^{0} ([0, 1])

is nonnegative, may be null in some set, but is positive in

[α, β]

.…”

Section: Introductionmentioning

confidence: 99%

x \in Ω

(as in ).…”

Section: Introductionmentioning

confidence: 99%

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Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros

Iturriaga

Massa

Sánchez

et al. 2014

Mathematische Nachrichten

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Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros

2021

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We study the nonlocal nonlinear problem $$\begin{aligned} \left\{ \begin{array}[c]{lll} (-\Delta )^s u = \lambda f(u) &{} \text{ in } \Omega , \\ u=0&{}\text{ on } \mathbb {R}^N{\setminus }\Omega , \quad (P_{\lambda }) \end{array} \right. \end{aligned}$$ ( - Δ ) s u = λ f ( u ) in Ω , u = 0 on R N \ Ω , ( P λ ) where $$\Omega $$ Ω is a bounded smooth domain in $$\mathbb {R}^N$$ R N , $$N>2s$$ N > 2 s , $$0<s<1$$ 0 < s < 1 ; $$f:\mathbb {R}\rightarrow [0,\infty )$$ f : R → [ 0 , ∞ ) is a nonlinear continuous function such that $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 and $$f(t)\sim |t|^{p-1}t$$ f ( t ) ∼ | t | p - 1 t as $$t\rightarrow 0^+$$ t → 0 + , with $$2<p+1<2^*_s$$ 2 < p + 1 < 2 s ∗ ; and $$\lambda $$ λ is a positive parameter. We prove the existence of two nontrivial solutions $$u_{\lambda }$$ u λ and $$v_{\lambda }$$ v λ to ($$P_{\lambda }$$ P λ ) such that $$0\le u_{\lambda }< v_{\lambda }\le 1$$ 0 ≤ u λ < v λ ≤ 1 for all sufficiently large $$\lambda $$ λ . The first solution $$u_{\lambda }$$ u λ is obtained by applying the Mountain Pass Theorem, whereas the second, $$v_{\lambda }$$ v λ , via the sub- and super-solution method. We point out that our results hold regardless of the behavior of the nonlinearity f at infinity. In addition, we obtain that these solutions belong to $$L^{\infty }(\Omega )$$ L ∞ ( Ω ) .

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Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros

2011

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Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros

Cited by 15 publications

References 21 publications

Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros

Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros

Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros

Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros

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