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Compactness results and applications to some “zero mass” elliptic problems. (English) Zbl 1159.35022

In this paper the existence of solutions for the semilinear elliptic equation \(-\Delta u=f'(u)\) in an unbounded subdomain of \(\mathbb{R}^3\) is considered. Specifically, the “zero mass” case is treated, that is, \(f'(0)\) and \(f''(0)\) vanish.
To use the variational structure of this class of problems, the natural space to work in is \(\mathcal{D}^{1,2}(\Omega)\), where \(\Omega\subseteq\mathbb{R}^3\) is unbounded. One also considers the space sum \(L^p+L^q(\Omega)\) with the norm
\[ \| v\| _{L^p+L^q(\Omega)}:=\inf\{\| v_1\| _{L^p(\Omega)}+\| v_2\| _{L^q(\Omega)}\mid (v_1,v_2)\in L^p(\Omega)\times L^q(\Omega),\;v=v_1+v_2\}. \]
If \(1<p<6<q\) then it is known that there is a continuous embedding \(\mathcal{D}^{1,2}(\Omega) \hookrightarrow L^p+L^q(\Omega)\).
Let \(\Omega\) have suitable symmetries and denote by \(\mathcal{D}_{\text{s}}^{1,2}(\Omega)\) a subspace of suitably symmetric functions of \(\mathcal{D}^{1,2}(\Omega)\). One of the aims of the authors is to prove the compactness of the embedding \(\mathcal{D}_{\text{s}}^{1,2}(\Omega) \hookrightarrow L^p+L^q(\Omega)\) in various situations. This information can be used to prove the relative compactness of Palais-Smale sequences for the application of variational methods.
In the first application, consider \(f\in C^1(\mathbb{C},\mathbb{R})\), satisfying \(f(0)=0\), \(f(M)>0\) for some \(M>0\), \(| f'(\xi)| \leq C\min\{| \xi| ^{p-1},| \xi| ^{q-1}\}\) for some constant \(C>0\) and all \(\xi\in\mathbb{C}\), and \(f(\xi) = f(| \xi| )\) for all \(\xi\in\mathbb{C}\). In particular, \(f\) has supercritical growth at \(0\) and subcritical growth at \(\infty\). It is proved that then for every \(n\in\mathbb{Z}\) the equation \(-\Delta v=f'(v)\) has a complex valued solution \(v^{(n)}\in\mathcal{D}^{1,2}(\mathbb{R}^3)\) of the form \(v^{(n)}(x,y,z)=u^{(n)}(r,z)\text{e}^{\text{i}n\theta}\), where \((r,\theta,z)\) represent cylindrical coordinates and \(u^{(n)}(r,z)\in\mathbb{R}\).
The second application is the equation \(-\Delta v=f'(v)\) posed on \(\mathbb{R}^2\times I\), where \(I\) is an open bounded interval in \(\mathbb{R}\). Here \(f\in C^1(\mathbb{R},\mathbb{R})\) satisfies \(f(0)=0\), \(f(\xi)\geq C_1\min\{| \xi| ^p,| \xi| ^q\}\) and \(f'(\xi)\leq C_2\min\{| \xi| ^{p-1},| \xi| ^{q-1}\}\) for some positive constants \(C_1,C_2\) and all \(\xi\in\mathbb{R}\), and the weak Ambrosetti-Rabinowitz condition \(\alpha f(\xi)\leq f'(\xi)\xi\) with some constant \(\alpha\geq2\) and for all \(\xi\in\mathbb{R}\). Then existence of infinitely many cylindrically symmetric solutions is proved.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:

[1] Almgren, F.J.; Lieb, E.H., Symmetric decreasing rearrangement is sometimes continuous, J. amer. math. soc., 2, 683-773, (1989) · Zbl 0688.46014
[2] Ambrosetti, A.; Felli, V.; Malchiodi, A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. eur. math. soc. (JEMS), 7, 117-144, (2005) · Zbl 1064.35175
[3] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[4] Aubin, T., Problémes isopérimétriques et espaces de Sobolev, J. differential geom., 11, 573-598, (1976) · Zbl 0371.46011
[5] A. Azzollini, A multiplicity result for a semilinear Maxwell type equation, preprint
[6] Azzollini, A.; Benci, V.; D’Aprile, T.; Fortunato, D., Existence of static solutions of the semilinear Maxwell equations, Ricerche mat., 55, 123-137, (2006)
[7] Badiale, M.; Rolando, S., Elliptic problems with singular potential and double-power nonlinearity, Mediterr. J. math., 4, 417-436, (2005) · Zbl 1121.35050
[8] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, J. nonlinear anal. TMA, 7, 981-1012, (1983) · Zbl 0522.58012
[9] Benci, V.; Fortunato, D., Towards a unified theory for classical electrodynamics, Arch. ration. mech. anal., 173, 379-414, (2004) · Zbl 1065.78004
[10] Benci, V.; Micheletti, A.M., Solutions in exterior domains of null mass nonlinear field equations, Adv. nonlinear stud., 6, 171-198, (2006) · Zbl 1114.35143
[11] Benci, V.; Grisanti, C.R.; Micheletti, A.M., Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with \(V(\infty) = 0\), Topol. methods nonlinear anal., 26, 203-219, (2005) · Zbl 1105.35112
[12] Benci, V.; Grisanti, C.R.; Micheletti, A.M., Existence of solutions for the nonlinear Schrödinger equations with \(V(\infty) = 0\), (), 53-65 · Zbl 1231.35225
[13] Benci, V.; Visciglia, N., Solitary wave with non-vanishing angular momentum, Adv. nonlinear stud., 3, 151-161, (2003) · Zbl 1030.35051
[14] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations. I. existence of a ground state, Arch. ration. mech. anal., 82, 313-345, (1983) · Zbl 0533.35029
[15] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations. II. existence of infinitely many solutions, Arch. ration. mech. anal., 82, 347-375, (1983) · Zbl 0556.35046
[16] Berestycki, H.; Lions, P.L., Existence d’états multiples dans des équations de champs scalaires non linéaires dans le cas de masse nulle, C. R. acad. sci. Paris Sér. I math., 297, 267-270, (1983) · Zbl 0542.35072
[17] Caffarelli, L.A.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. pure appl. math., 42, 3, 271-297, (1989) · Zbl 0702.35085
[18] Esteban, M.J.; Lions, P.L., A compactness lemma, Nonlinear anal. TMA, 7, 381-385, (1983) · Zbl 0512.46035
[19] M. Ghimenti, A.M. Micheletti, Existence of minimal nodal solutions for the nonlinear Schroedinger equations with \(V(\infty) = 0\), preprint · Zbl 1146.35412
[20] Krasnosel’skii, M.A.; Rutickii, Y.B., Convex functions and Orlicz spaces, (1961), P. Noordhoff Ltd Groningen
[21] Lions, P.L., Minimization problems in \(L^1(\mathbb{R}^3)\), J. funct. anal., 41, 236-275, (1981) · Zbl 0464.49019
[22] Lions, P.L., Symétrie et compacité dans LES espaces de Sobolev, J. funct. anal., 49, 315-334, (1982) · Zbl 0501.46032
[23] Lions, P.L., Solutions complexes d’équations elliptiques semilinéaires dans \(\mathbb{R}^N\), C. R. acad. sci. Paris S. I math., 302, 673-676, (1986) · Zbl 0606.35027
[24] L. Pisani, Remarks on the sum of Lebesgue spaces, preprint
[25] Talenti, G., Best constant in Sobolev inequality, Ann. mat. pura appl., 110, 353-372, (1976) · Zbl 0353.46018
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