Positive solutions of asymptotically linear equations via Pohozaev manifold

Lehrer, Raquel; Maia, Liliane A.

doi:10.1016/j.jfa.2013.09.002

Cited by 53 publications

(44 citation statements)

References 23 publications

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“…This is the main reason why we are led to look for a different approach, using the Pohozaev identity and manifold. De Figueiredo, Lions and Nussbaum in [11], Lemma 1.1, (see also Proposition 2.1 in [16]) provide a Pohozaev type identity for problem (P ). We observe that from Lemma 1.1 in [11], any solution of equation (P ) satisfies the Pohozaev identity…”

Section: A Pohozaev Type Identitymentioning

confidence: 87%

“…Therefore, one is motivated to use the more suitable projections on the set of points which satisfy the Pohozaev identity [19], the so-called the Pohozaev manifold of (P ), instead. In recent years the use of Pohozaev manifold was shown very effective when treating nonlinearities which do not satisfy Ambrosetti-Rabinowitz superquadraticity condition [2] and the monotonicity condition f (s)/s increasing [12]; this idea is performed in [3,16,18] and [23] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, if f (x, s)/s is not increasing of s on [0, ∞) then the projections on Nehari manifold are no longer suitable. Our purpose is to face this problem under this generality adapting the ideas developed in [16]. Indeed, this is the counterpart of the problem in [16] where the variable potential plays the role of the nonautonomous terms in there.…”

Section: Introductionmentioning

confidence: 99%

“…Our purpose is to face this problem under this generality adapting the ideas developed in [16]. Indeed, this is the counterpart of the problem in [16] where the variable potential plays the role of the nonautonomous terms in there. Some of the arguments performed here are closely related to those of our previous paper [16].…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, this is the counterpart of the problem in [16] where the variable potential plays the role of the nonautonomous terms in there. Some of the arguments performed here are closely related to those of our previous paper [16]. So, in the proofs of some technical results we will omit the details and refer to where adaptations are needed.…”

Section: Introductionmentioning

confidence: 99%

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Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential

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Section: A Pohozaev Type Identitymentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential

Lehrer

Maia

Ruviaro

2014

Nonlinear Differ. Equ. Appl.

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On the Neumann problem for fractional semilinear elliptic equations arising from Keller–Segel model

Jin

Sun

2022

Math Methods in App Sciences

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In this paper, we study the following fractional semilinear Neumann problem arising from Keller–Segel model: {array(−ϵΔ)1/2u+u=h(u)arrayinΩ,array∂νu=0arrayon∂Ω,$$ \left\{\begin{array}{cc}{\left(-\epsilon \Delta \right)}^{1/2}u+u=h(u)\kern0.60em & \kern0.1em \mathrm{in}\kern0.5em \Omega, \\ {}{\partial}_{\nu u}=0\kern0.30em & \mathrm{on}\kern0.5em \mathrm{\partial \Omega },\end{array}\right. $$ where normalΩ⊂ℝn0.1emfalse(n≥2false)$$ \Omega \subset {\mathbb{R}}^n\kern0.1em \left(n\ge 2\right) $$ is a smooth bounded domain and ν$$ \nu $$ is the outward unit normal to ∂normalΩ$$ \mathrm{\partial \Omega } $$. First, under the superlinear and subcritical growth assumptions on h$$ h $$, we prove that there exists at least one positive nonconstant solution uϵ$$ {u}_{\epsilon } $$ for small ϵ>0$$ \epsilon >0 $$ and the family of solutions false{uϵfalse}ϵ>0$$ {\left\{{u}_{\epsilon}\right\}}_{\epsilon >0} $$ is uniformly bounded. Moreover, we build a Pohozaev‐type identity for the ( ϵ$$ \epsilon $$‐) Neumann harmonic extension of the following problem: {array(−ϵΔ)1/2u=g(u)arrayinΩ,array∂νu=0arrayon∂Ω,$$ \left\{\begin{array}{cc}{\left(-\epsilon \Delta \right)}^{1/2}u=g(u)\kern0.60em & \kern0.1em \mathrm{in}\kern0.5em \Omega, \\ {}{\partial}_{\nu u}=0\kern0.30em & \mathrm{on}\kern0.5em \mathrm{\partial \Omega },\end{array}\right. $$ where g$$ g $$ is a C1$$ {C}^1 $$ function such that gfalse(0false)=0$$ g(0)=0 $$. As a direct application of this identity, when g$$ g $$ satisfies false(n−1false)tgfalse(tfalse)−2nGfalse(tfalse)≥0$$ \left(n-1\right) tg(t)-2 nG(t)\ge 0 $$, where Gfalse(tfalse)=∫0tgfalse(sfalse)ds$$ G(t)={\int}_0^tg(s) ds $$, we deduce the nonexistence of weak bounded solution in star‐shaped domains.

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Existence and Nonexistence Results for a Schrödinger Equation with Saturable Nonlinearity

Lehrer

Maia

2014

Progress in Nonlinear Differential Equations and Their Applications

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Positive solutions of asymptotically linear equations via Pohozaev manifold

Cited by 53 publications

References 23 publications

Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential

Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential

On the Neumann problem for fractional semilinear elliptic equations arising from Keller–Segel model

Existence and Nonexistence Results for a Schrödinger Equation with Saturable Nonlinearity

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