Multiple positive solutions for a class of concave–convex elliptic problems in ℝNinvolving sign-changing weight, II

Huang, Yisheng; Wu, Tsung‐fang; Wu, Yuanze

doi:10.1142/s021919971450045x

Cited by 24 publications

(15 citation statements)

References 26 publications

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“…Since (P λ1 ) has no solution, it follows from the strong maximum principle that u 0 = 0 in H 1 0 (B R ). Note that α n ↑ λ 1 , we can see from similar arguments used in the proof of [12,Lemma 5.2] that (iii) 0 < λ < aλ 1 and µ > bS 2 in the case N = 4;…”

Section: 3mentioning

confidence: 56%

“…Furthermore, a similar argument used in the proof of [12,Lemma 5.2] shows that m α < m 0 for all α ∈ (0, λ 1 ). It follows from the Hölder inequality that…”

Section: The Case Of 2 = Q < P < 2 *mentioning

confidence: 76%

“…We first prove that λ 0 ≤ 0. Suppose on the contrary, then by a similar argument as used in [12,Lemma 5.2], we can see that m α < m 0 for all α ∈ (λ 0 , λ 1 ). It follows that u α and α ∈ R}, then U is an open set in L 2 (Ω) × R, which contains the point (0, λ 1 ).…”

Section: Lemma 52mentioning

confidence: 77%

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On finding solutions of a Kirchhoff type problem

Huang

Liu

2015

Proc. Amer. Math. Soc.

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Consider the following Kirchhoff type problemwhereN −2 and a, b, λ, µ are positive parameters. By introducing some new ideas and using the well-known results of the problem (P) in the cases of a = µ = 1 and b = 0, we obtain some special kinds of solutions to (P) for all N ≥ 3 with precise expressions on the parameters a, b, λ, µ, which reveals some new phenomenons of the solutions to the problem (P). It is also worth to point out that it seems to be the first time that the solutions of (P) can be expressed precisely on the parameters a, b, λ, µ, and our results in dimension four also give a partial answer to Neimen's open problems

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Section: 3mentioning

confidence: 56%

“…Furthermore, a similar argument used in the proof of [12,Lemma 5.2] shows that m α < m 0 for all α ∈ (0, λ 1 ). It follows from the Hölder inequality that…”

Section: The Case Of 2 = Q < P < 2 *mentioning

confidence: 76%

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On finding solutions of a Kirchhoff type problem

Huang

Liu

2015

Proc. Amer. Math. Soc.

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“…By the variational method, they obtained the existence and multiplicity of positive solutions to the above problem. Subsequently, an increasing number of researchers have paid attention to semilinear elliptic equations with critical exponent and concave-convex nonlinearities; for example, see [ 1 , 5 , 13 , 14 , 27 , 29 ] and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy–Sobolev exponent and singular nonlinearity

Shen

2018

J Inequal Appl

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In this paper, we study a class of critical elliptic problems of Kirchhoff type: where , , , and are constants and is the Hardy–Sobolev exponent in . For a suitable function , we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard as a parameter to obtain the convergence property of solutions for the given problem as by the mountain pass theorem and Ekeland’s variational principle.

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“…Here we cite the pioneer work [2] where several results are proved on bounded domains Ω ⊂ R N . In the whole space concave-convex nonlinearities have been considered assuming extra assumptions on the potential V , see [8,13,34]. Another contribution in this work is to consider the nonlinear Rayleigh quotient proving existence of a parameter λ * > 0 such that Problem (1.1) admits at least two solutions for each λ ∈ (0, λ * ].…”

mentioning

confidence: 99%

Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities

Carvalho¹,

Silva²,

Goulart³

2021

CPAA

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It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u +V(x) u = (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u \, {\rm{\;in\;}}\, \mathbb{R}^N, \\ \ u\in H^1( \mathbb{R}^N) \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M3">\begin{document}$ \lambda > 0, N \geq 3, \alpha \in (0, N) $\end{document}</tex-math></inline-formula>. The potential <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a continuous function and <inline-formula><tex-math id="M5">\begin{document}$ I_\alpha $\end{document}</tex-math></inline-formula> denotes the standard Riesz potential. Assume also that <inline-formula><tex-math id="M6">\begin{document}$ 1 < q < 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ 2_\alpha </inline-formula> where <inline-formula><tex-math id="M8">\begin{document}$ 2_\alpha = (N+\alpha)/N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ 2_\alpha = (N+\alpha)/(N-2) $\end{document}</tex-math></inline-formula>. Our main contribution is to consider a specific condition on the parameter <inline-formula><tex-math id="M10">\begin{document}$ \lambda > 0 $\end{document}</tex-math></inline-formula> taking into account the nonlinear Rayleigh quotient. More precisely, there exists <inline-formula><tex-math id="M11">\begin{document}$ \lambda^* > 0 $\end{document}</tex-math></inline-formula> such that our main problem admits at least two positive solutions for each <inline-formula><tex-math id="M12">\begin{document}$ \lambda \in (0, \lambda^*] $\end{document}</tex-math></inline-formula>. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter <inline-formula><tex-math id="M13">\begin{document}$ \lambda^*> 0 $\end{document}</tex-math></inline-formula> is optimal in some sense which allow us to apply the Nehari method.

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Multiple positive solutions for a class of concave–convex elliptic problems in ℝ^Ninvolving sign-changing weight, II

Cited by 24 publications

References 26 publications

On finding solutions of a Kirchhoff type problem

On finding solutions of a Kirchhoff type problem

Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy–Sobolev exponent and singular nonlinearity

Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities

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