Positive weak solutions of a generalized supercritical Neumann problem
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RESEARCH PAPER
Positive weak solutions of a generalized supercritical Neumann problem F. Safari1 • A. Razani1 Received: 5 July 2020 / Accepted: 23 September 2020 Shiraz University 2020
Abstract The aim of this paper is to show that a supercritical Neumann problem has at least one positive radial solution via recent variational principle and defining an energy functional with a hidden symmetry associated with the problem. Also, we study the problem in the setting of the Heisenberg group Hn , which is a special case of locally compact groups. Keywords Variational principle Supercritical Neumann problem Mountain pass geometry theorem Mathematics Subject Classification 35J15 58E30
1 Introduction PDE is one of the interdisciplinary branches of mathematics that can give a suitable answer to engineering, physical and social science problems. Thus, researches promote this acknowledge to solve problems. There are many articles in these topics, and one can see Ragusa and Tachikawa (2020) and Razani (2002a, b, 2004a, b, 2007, 2014, 2018, 2019) for more relevant problems in PDEs. There are many techniques to study the problems, and one of the techniques to variational methods dates back to 1969 when the brachistochrone problem was posed and solved by Newton, Leibniz, Jakob and Bernoulli. In 1744, Euler published the first monograph on the calculus of variations and Lagrange introduced formalisms and techniques. Lagrange’s method is still used in solving physical problems (see, for some recent articles, Abdolrazaghi and Razani (2019, 2020); Behboudi et al. (2020); Behboudi and Razani (2019); Ragusa and Razani (2020); Safari and
& A. Razani [email protected] F. Safari [email protected] 1
Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, P.O. Box 34194-168181, Iran
Razani (2020, 2020a, b); Makvand Chaharlang et al. (2020); and Makvand Chaharlang and Razani (2020). In fact, not for solving all differential problems (see Aghajani et al. (2018a, b) and Cowan and Razani (2020a, b) for nonvariational equation) but for a wide class of them, a functional known as Euler–Lagrange energy functional associated with the problem is being defined where its critical points are solutions of the problem. Due to this, we review some papers as follows: Ni (1982) proved that He`non equation Du ¼ jxja up x 2 B; u¼0 x 2 oB; has a positive solution if and only if p\ nþ2þ2a n2 , where a [ 0 and B is the unit ball in Rn centered at the origin. Lerman et al. (2020) considered Du þ u ¼ u3 : in R2 and proved the existence of its periodic solutions and generalized their results for higher dimensions. Serra and Tilli (2011) studied the problem 8 Du þ u ¼ aðjxjÞf ðuÞ x 2 B; > > < u[0 x 2 B; > ou > : ¼0 x 2 oB; on where f(u) is a supercritical nonlinear function and a is a radial function. To verify the existence of solution for this
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problem, they obtained the critical points of the associated classical
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