Kirchhoff elliptic problems with asymptotically linear or superlinear nonlinearities
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Kirchhoff elliptic problems with asymptotically linear or superlinear nonlinearities Marcelo F. Furtado , Edcarlos D. Silva and Uberlandio B. Severo Abstract. We establish the existence and multiplicity of solutions for Kirchhoff elliptic problems of type ⎞ ⎛ 2 −m ⎝ |∇u| dx⎠ Δu = f (x, u), x ∈ R3 , R3
where m : R+ → R is continuous, positive and satisfies appropriate growth and/or monotonicity conditions. We consider the cases that f is asymptotically 3−linear or 3−superlinear at infinity, in an appropriated sense. By using variational methods, we obtain our results under crossing assumptions of the functions m and f with respect to limit eigenvalues problems. In the model case m(t) = a + bt, we also prove a concentration result for some solutions when b → 0+ . Mathematics Subject Classification. Primary 35J50, Secondary 47G20. Keywords. Kirchhoff equation, Asymptotically linear problems, Superlinear problems, Variational methods.
1. Introduction Consider the problem
⎛
−m ⎝
⎞ |∇u|2 dx⎠ Δu = g(x, u)
in
Ω,
u = 0 on
∂Ω,
Ω
where Ω ⊂ RN is a bounded smooth domain, m : R+ → R is a positive function and the function nonlinear |∇u|2 dx in the g has polynomial growth. It is called nonlocal due to the presence of the term m Ω
equation, and it has its origin in the theory of nonlinear vibrations. For instance, in the case m(t) = a+bt, with a, b > 0, it comes from the following model for the modified d’Alembert wave equation ⎛ ⎞ L 2 2
∂u
E ∂ 2 u ⎝ P0
dx⎠ ∂ u = g(x, u), + ρ 2 − ∂t h 2L ∂x
∂x2 0
for free vibrations of elastic strings. Here, L is the length of the string, h is the area of the cross section, E is the Young modulus of the material, ρ is the mass density and P0 is the initial tension. This kind of nonlocal equation was first proposed by Kirchhoff [20], and it was considered theoretically or experimentally by several physicists after that (see [11,24,25,27]). Nonlocal problems also appear in other fields as, for The first author was partially supported by CNPq/Brazil Grant 308673/2016-6 and FAPDF/Brazil. The second author was partially supported by CNPq Grant 429955/2018-9. The third author was partially supported by CNPq Grant 310747/2019-8. 0123456789().: V,-vol
186
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M. F. Furtado, E. D. Silva and U. B. Severo
ZAMP
example, biological systems where u describes a process which depends on the average of itself (for instance, population density). We refer the reader to [16,23], and references therein, for more examples on the physical motivation of this problem. In the present work, we are interested in the case that the problem is settled in the entire space R3 . More specifically, we consider
−m u2 Δu = f (x, u), x ∈ R3 , (P ) u ∈ D1,2 (R3 ), where u = ( |∇u|2 dx)1/2 and D1,2 (R3 ) is the closure of C0∞ (R3 ) with respect to the norm · . We R3
have some structural assumptions on m ∈ C(R), and we shall consider two different classes of functions f ∈ C(R3 × R) depending on the growth at infinity: the asymptoticall
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