Ground States for Elliptic Equations in \(\pmb{\mathbb{R}}^{2}\) with Exponential Critical Growth
In this paper, we obtain a mountain pass characterization of ground state solutions for some class of elliptic equations in ℝ2 with nonlinearities in the critical (exponential) growth range.
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Abstract In this paper, we obtain a mountain pass characterization of ground state solutions for some class of elliptic equations in R2 with nonlinearities in the critical (exponential) growth range. Keywords Ground state solutions · Elliptic equations in R2 · Exponential critical growth · Variational methods
1 Introduction This paper is concerned with the existence of solutions of a nonlinear scalar field equation of the form (1) −Δu = g(u) in R2 , u ∈ H 1 R2 , and in particular we will study the following problem −Δu + u = f (u)
in R2 , u ∈ H 1 R2 ,
(2)
that is, problem (1) with g(s) := f (s) − s. The study of these kind of problems is motivated by applications in many areas of mathematical physics. In particular, solutions of (2) provide stationary states for the nonlinear Klein-Gordon equation and for the nonlinear Schrödinger equation. Problem (1) has been extensively studied starting from the fundamental papers due to Berestycki and Lions [4] and to Berestycki, Gallouët and Kavian [5]. We recall that these papers are both concerned with subcritical nonlinearities, in particular in [4] the authors treated nonlinearities with subcritical polynomial growth, while in [5] the authors treated nonlinearities with subcritical exponential growth. From now on, we will focus our attention on the case when the nonlinear term is of exponential B. Ruf (B) · F. Sani Dipartimento di Matematica Federigo Enriques, Università degli Studi di Milano, via Saldini 50, Milano, Italy e-mail: [email protected] F. Sani e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_16, © Springer-Verlag Italia 2013
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type, since our aim is to study problem (2) with a nonlinearity f exhibiting a critical exponential growth. The maximal growth which can be treated variationally in the Sobolev space H 1 (R2 ) is given by the Trudinger-Moser inequality: Theorem 1 [10, Theorem 1.1] There exists a constant C > 0 such that 4πu2 sup e − 1 dx ≤ C u∈H 1 (R2 ),uH 1 ≤1 R2
(3)
where u2H 1 := ∇u22 + u22 is the standard Sobolev norm. This inequality is sharp: if we replace the exponent 4π with any α > 4π the supremum is infinite. In view of this inequality we say that a nonlinearity f has critical growth if there exists α0 > 0 such that 0 for α > α0 , |f (s)| lim = 2 |s|→+∞ eαs +∞ for α < α0 . Our aim is to obtain a mountain pass characterization of ground state solutions of problem (2). The natural functional corresponding to a variational approach to problem (2) is 1 I (u) := F (u) dx |∇u|2 + u2 dx − 2 R2 R2 1 = |∇u|2 dx − G(u) dx, u ∈ H 1 R2 , 2 R2 R2 s s where F (s) := 0 f (t) dt and G(s) := 0 g(t) dt. We will say that I has a mountain pass geometry, if the following conditions hold: (I0 ) I (0) = 0; (I1 ) there exist ρ, a > 0 such that I (u) ≥ a > 0 for any u ∈ H 1 (R2 ) with uH 1 = ρ; (I2 ) there exists u0 ∈ H 1 (R2 ) such that u0 H 1 > ρ and I (u0 ) < 0. W
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