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The Schrödinger-Poisson equation under the effect of a nonlinear local term. (English) Zbl 1136.35037

The main object of the paper is the Schrödinger-Poisson problem
\[ -\Delta u+u+ \lambda\varphi u= u^p, \qquad -\Delta\varphi= u^2, \qquad \lim_{|x|\to\infty}\varphi(x)=0, \]
where \(u,\varphi:\mathbb R^3\to\mathbb R\) are positive radial functions, \(\lambda>0\) and \(1<p<5\). Its solutions coincide with critical points of the associated energy functional
\[ I_\lambda(u)= \int\frac12 \left(|\nabla u(x)|^2+ u^2(x))+ \frac14 \varphi_u(x) u^2(x)- \frac{1}{p+1} |u(x)|^{p+1} \right)\,dx, \] considered for \(u\) from the Sobolev spaces \(H_r^1\).
Existence and nonexistence results are contained in the following table:
\[ \begin{matrix}\l&\qquad\l&\qquad\l&\qquad\l&\qquad\l\\ &\lambda\text{ small} &&\lambda\geq 1/4&{}\\ 1<p<2 &\text{Two solutions} &\inf I>-\infty &\text{No solutions} &\inf I=0 \\ p=2 &\text{One solution} &\inf I=-\infty &\text{No solutions} &\inf I=0 \\ 2<p<5 &\text{One solution} &\inf I=-\infty &\text{One solution} &\inf I=-\infty \end{matrix} \]
i.e., \(p=2\) is the article value for the existence of solutions

MSC:

35J60 Nonlinear elliptic equations
35J50 Variational methods for elliptic systems
47J30 Variational methods involving nonlinear operators
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[1] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401
[2] Ambrosetti, A.; Malchiodi, A., Perturbation methods and semilinear elliptic problems on \(\mathbb{R}^n\), Progr. math., vol. 240, (2005), Birkhäuser Boston Boston, MA
[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] Benci, V.; Fortunato, D., An eigenvalue problem for the schrödinger – maxwell equations, Topol. methods nonlinear anal., 11, 283-293, (1998) · Zbl 0926.35125
[5] Benci, V.; Fortunato, D., Solitary waves of the nonlinear klein – gordon equation coupled with Maxwell equations, Rev. math. phys., 14, 409-420, (2002) · Zbl 1037.35075
[6] Benguria, R.; Brezis, H.; Lieb, E.-H., The thomas – fermi – von Weizsäcker theory of atoms and molecules, Comm. math. phys., 79, 167-180, (1981) · Zbl 0478.49035
[7] Catto, I.; Lions, P.-L., Binding of atoms and stability of molecules in Hartree and thomas – fermi type theories. part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. partial differential equations, 17, 1051-1110, (1992) · Zbl 0767.35065
[8] I. Catto, O. Sánchez, J. Soler, Nondispersive dynamics stemming from \(X^\alpha\) corrections to the Schrödinger-Poisson systems, preprint
[9] Chang, K.-C., Infinite-dimensional Morse theory and multiple solutions problems, (1993), Birkhäuser Basel
[10] Coclite, G.M., A multiplicity result for the nonlinear schrödinger – maxwell equations, Commun. appl. anal., 7, 417-423, (2003) · Zbl 1085.81510
[11] T. D’Aprile, Semiclassical states for the nonlinear Schrödinger equation with the electromagnetic field, preprint
[12] D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear klein – gordon – maxwell and schrödinger – maxwell equations, Proc. roy. soc. Edinburgh sect. A, 134, 893-906, (2004) · Zbl 1064.35182
[13] D’Aprile, T.; Mugnai, D., Non-existence results for the coupled klein – gordon – maxwell equations, Adv. nonlinear stud., 4, 307-322, (2004) · Zbl 1142.35406
[14] T. D’Aprile, J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, preprint
[15] D’Avenia, P., Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. nonlinear stud., 2, 177-192, (2002) · Zbl 1007.35090
[16] Ekeland, I., On the variational principle, J. math. anal. appl., 47, 324-353, (1974) · Zbl 0286.49015
[17] Gidas, B.; Ni, W.-M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbb{R}^n\), (), 369-402
[18] Kwong, M.K., Uniqueness of positive solutions of \(\operatorname{\Delta} u - u + u^p = 0\) in \(\mathbb{R}^n\), Arch. ration. mech. anal., 105, 243-266, (1989) · Zbl 0676.35032
[19] Lieb, E.H., Thomas – fermi and related theories and molecules, Rev. modern phys., 53, 603-641, (1981) · Zbl 1049.81679
[20] Lieb, E.H.; Simon, B., The thomas – fermi theory of atoms, molecules and solids, Adv. math., 23, 22-116, (1977) · Zbl 0938.81568
[21] Lions, P.-L., Solutions of hartree – fock equations for Coulomb systems, Comm. math. phys., 109, 33-97, (1984) · Zbl 0618.35111
[22] Markowich, P.; Ringhofer, C.; Schmeiser, C., Semiconductor equations, (1990), Springer-Verlag New York · Zbl 0765.35001
[23] Oh, Y.-G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials, Comm. math. phys., 131, 223-253, (1990) · Zbl 0753.35097
[24] Ruiz, D., Semiclassical states for coupled schrödinger – maxwell equations: concentration around a sphere, Math. model. methods appl. sci., 15, 141-164, (2005) · Zbl 1074.81023
[25] Sánchez, O.; Soler, J., Long-time dynamics of the schrödinger – poisson – slater system, J. statist. phys., 114, 179-204, (2004) · Zbl 1060.82039
[26] Strauss, W.A., Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028
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