## 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
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.