Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity

• Echarghaoui, Rachid ^[1] ; Sersif, Rachid ^[1] ; Zaimi, Zakaria ^[1]
  1. [1] Department of Mathematics, Faculty of Sciences, Ibn Tofail University, B. P. 133, Kenitra, Morocco
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 1, 2025, págs. 163-185
• Idioma: inglés
• DOI: 10.1007/s13348-023-00426-4
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• Resumen

  • The authors of Cao and Yan (J Differ Equ 251:1389–1414, 2011) have considered the following semilinear critical Neumann problem:

    \begin{aligned} \varvec{-\Delta u=\vert u\vert ^{2^{*}-2} u+g(u) \quad \text{ in } \Omega , \quad \frac{\partial u}{\partial \nu }=0 \quad \text{ on } \partial \Omega ,} \end{aligned} where \varvec{\Omega } is a bounded domain in \varvec{\mathbb {R}^{N}} satisfying some geometric conditions, \varvec{\nu } is the outward unit normal of \varvec{\partial \Omega , 2^{*}:=\frac{2 N}{N-2}} and \varvec{g(t):=\mu \vert t\vert ^{p-2} t-t,} where \varvec{p \in \left( 2,2^{*}\right) } and \varvec{\mu >0} are constants. They proved the existence of infinitely many solutions with positive energy for the above problem if \varvec{N>\max \left( \frac{2(p+1)}{p-1}, 4\right) .} In this present paper, we consider the case where the exponent \varvec{p \in \left( 1,2\right) } and we show that if \varvec{N>\frac{2(p+1)}{p-1},} then the above problem admits an infinite set of solutions with positive energy. Our main result extend that obtained by P. Han in [9] for the case of elliptic problem with Dirichlet boundary conditions.

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