Nontrivial solutions to non-local problems with sublinear or superlinear nonlinearities
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ORIGINAL PAPER
Nontrivial solutions to non-local problems with sublinear or superlinear nonlinearities Zhiqing Han1
•
Ye Xue1
Received: 22 March 2020 / Accepted: 31 July 2020 Ó Springer Nature Switzerland AG 2020
Abstract Existence of nontrivial and multiple solutions for two types of non-local problems with sublinear or superlinear nonlinearities are investigated by linking theorems and index theory in critical point theory. Some results in the literature are extended. Keywords Non-local(Fractional) problems Nontrivial(Multiple) solutions Linking theorems Index theory Mathematics Subject Classification 35R11 35A15 47A10
1 Introduction Fractional and non-local operators of elliptic type arise in a quite natural way in many different problems, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering,minimal surfaces, materials science, water waves and so on. The investigations of the problems involved these non-local operators are interesting and important from both pure mathematical research aspects and real-world applications, eg see [1, 2] and references therein. Recently, variational methods and critical point theory have been proved to be powerful in dealing with these non-local elliptic problems after the paper [3] establishing the framework for the solvability of the following problems This article is part of the topical collection dedicated to Prof. Dajun Guo for his 85th birthday, edited by Yihong Du, Zhaoli Liu, Xingbin Pan, and Zhitao Zhang. & Zhiqing Han [email protected] Ye Xue [email protected] 1
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China SN Partial Differential Equations and Applications
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SN Partial Differ. Equ. Appl. (2020)1:29
ðDÞs u ¼ gðx; uÞ
in
X;
in RN nX;
u¼0
ð1Þ
where X RN is an open bounded set with smooth boundary, 0\s\1, N [ 2s and ðDÞs is the fractional Laplace operator, which (up to normalization factors) may be defined as Z ðuðx þ yÞ þ uðx yÞ 2uðxÞÞ ðDÞs uðxÞ ¼ dy; x 2 RN : jyjNþ2s RN They established the solvability of nontrivial solutions of the problems under the Ambrosetti- Rabinowitz superlinear condition for the nonlinearity g(x, u): there exist l [ 2 and r [ 0 such that for a.e. x 2 X; t 2 R; jtj r, we have 0\lGðx; tÞ tgðx; tÞ; Rt where G ¼ 0 gðx; sÞds. When the nonlinearity g(x, u) satisfies a linear growth condition, solvability of the problem is studied in [4]. When g(x, u) is a lower order perturbation of the critical power, the classical Brezis-Nirenberg results are established in [5]. Some multiplicity results are also established either by Morse theory eg see [6, 7] or by fountain theorems eg see [8] where superlinear nonlinearities without Ambrosetti- Rabino
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