Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter

Motreanu, Dumitru; Tanaka, Mieko

doi:10.1007/s10231-013-0327-9

Cited by 18 publications

(18 citation statements)

References 23 publications

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“…Since B may merely be continuous, we first construct a locally Lipschitz continuous operator A on

X = H_{0}^{1} false(normalΩ false) ∖ K

, which inherits the properties of B . Using the properties of B described in Lemmas 3.4 and 3.5, the next result follows from a similar argument as in [, Lemma 4.1; , Lemma 17]. Lemma Assume that hypothesis H 2 hold.…”

Section: Resonance Problems Withmentioning

confidence: 81%

“…Also, we consider the nonlinear map

T : H_{0}^{1} false(normalΩ false) \to H_{0}^{1} {(Ω)}^{★}

defined for all

u, v \in H_{0}^{1} false(normalΩ false)

0true (〈〉 T (u), v) = (〈〉 V_{p} false(u false), v) + (〈〉 V (u), v) + ξ_{ρ} \int_{Ω} u (z) v (z) normald z,

where

ξ_{ρ}

is given in hypothesis H 2 (iv), and

ρ = max {{false∥ u_{0} false∥}_{\infty}, {false∥ v_{0} false∥}_{\infty}}

(see Theorem 3.2). Then, the inverse

T^{- 1} : H_{0}^{1} false(normalΩ false)^{★} \to H_{0}^{1} false(normalΩ false)

of T exists and it is continuous (see [, Proposition 9]).…”

Section: Resonance Problems Withmentioning

confidence: 99%

“…Consider the map

T_{1} : W_{0}^{1, p} false(normalΩ false) \to W_{0}^{1, p} {(Ω)}^{★}

defined for all

u, v \in W_{0}^{1, p} false(normalΩ false)

0true (〈〉 T_{1} false(u false), v) = (〈〉 V_{p} false(u false), v) + (〈〉 V (u), v) + ξ_{ρ_{1}} \int_{Ω} {|u|}^{p - 2} u (z) v (z) normald z,

where

ξ_{ρ_{1}}

is given in hypothesis H 4 (vi). Then, the inverse

T_{1}^{- 1} : W_{0}^{1, p} false(normalΩ false)^{★} \to W_{0}^{1, p} false(normalΩ false)

of T 1 exists and it is continuous (see [, Proposition 9]). Let us define a map

A_{1} : W_{0}^{1, p} false(normalΩ false) \to W_{0}^{1, p} false(normalΩ false)

A_{1} false(u false) = T_{1}^{- 1} false(f (\cdot, u) + ξ_{ρ_{1}} {|u|}^{p - 2} u false), u \in W_{0}^{1, p} ...

…”

Section: Superlinear Problems Withmentioning

confidence: 99%

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Nodal solutions for resonant and superlinear (p, 2)‐equations

Lei

Zhang

et al. 2018

Mathematische Nachrichten

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We consider nonlinear, nonhomogeneous elliptic Dirichlet equations driven by the sum of a p‐Laplacian and a Laplacian (so‐called (p, 2)‐equation). We are concerned with both cases 12. In the first one, the reaction f(z,x) is linear grow near ±∞ and resonant with respect to a nonprincipal nonnegative eigenvalue. In the second case, the reaction f(z,·) is (p−1)‐superlinear near ±∞ and has z‐dependent zeros of constant sign. Using variational methods together with flow invariance arguments, we establish the existence of nodal solutions.

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“…Since B may merely be continuous, we first construct a locally Lipschitz continuous operator A on

X = H_{0}^{1} false(normalΩ false) ∖ K

Section: Resonance Problems Withmentioning

confidence: 81%

“…Also, we consider the nonlinear map

T : H_{0}^{1} false(normalΩ false) \to H_{0}^{1} {(Ω)}^{★}

defined for all

u, v \in H_{0}^{1} false(normalΩ false)

0true (〈〉 T (u), v) = (〈〉 V_{p} false(u false), v) + (〈〉 V (u), v) + ξ_{ρ} \int_{Ω} u (z) v (z) normald z,

where

ξ_{ρ}

is given in hypothesis H 2 (iv), and

ρ = max {{false∥ u_{0} false∥}_{\infty}, {false∥ v_{0} false∥}_{\infty}}

(see Theorem 3.2). Then, the inverse

T^{- 1} : H_{0}^{1} false(normalΩ false)^{★} \to H_{0}^{1} false(normalΩ false)

of T exists and it is continuous (see [, Proposition 9]).…”

Section: Resonance Problems Withmentioning

confidence: 99%

“…Consider the map

T_{1} : W_{0}^{1, p} false(normalΩ false) \to W_{0}^{1, p} {(Ω)}^{★}

defined for all

u, v \in W_{0}^{1, p} false(normalΩ false)

0true (〈〉 T_{1} false(u false), v) = (〈〉 V_{p} false(u false), v) + (〈〉 V (u), v) + ξ_{ρ_{1}} \int_{Ω} {|u|}^{p - 2} u (z) v (z) normald z,

where

ξ_{ρ_{1}}

is given in hypothesis H 4 (vi). Then, the inverse

T_{1}^{- 1} : W_{0}^{1, p} false(normalΩ false)^{★} \to W_{0}^{1, p} false(normalΩ false)

of T 1 exists and it is continuous (see [, Proposition 9]). Let us define a map

A_{1} : W_{0}^{1, p} false(normalΩ false) \to W_{0}^{1, p} false(normalΩ false)

A_{1} false(u false) = T_{1}^{- 1} false(f (\cdot, u) + ξ_{ρ_{1}} {|u|}^{p - 2} u false), u \in W_{0}^{1, p} ...

…”

Section: Superlinear Problems Withmentioning

confidence: 99%

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Nodal solutions for resonant and superlinear (p, 2)‐equations

Lei

Zhang

et al. 2018

Mathematische Nachrichten

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show abstract

“…Under Dirichlet boundary conditions, the above equation has been widely investigated; see for instance [21,4,19] and the references given there.…”

Section: Nodal Solutionsmentioning

confidence: 99%

“…(x, t) ∈ Ω × R, where λ > λ 2 , is also examined and some results of [22] extended; cf. also [5,11,19], which however require β ≡ 0. When p = 2 we obtain a second nodal solution by assuming, among other things, f (x, ·) ∈ C 1 (R) and…”

Section: Introductionmentioning

confidence: 99%