Abstract.
We consider a class of equations of the form \(-\varepsilon^2\Delta u + V(x)u = f(u), \quad u\in H^1({\bf R}^N).\) By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as \(\varepsilon\to 0\). We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function \(\xi\mapsto \frac{f(\xi)}{\xi}\). We deal with asymptotically linear as well as superlinear nonlinearities.
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References
Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140, 285-300 (1997)
Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I. Arch. Rat. Mech. Anal. 82, 313-346 (1983)
Berestycki, H., Gallouët, T., Kavian, O.: Equations de Champs scalaires euclidiens non lin\’ eaires dans le plan. C.R. Acad. Sci; Paris Ser. I Math. 297, 307-310 (1983) and Publications du Laboratoire d’Analyse Numérique, Université de Paris VI (1984)
Brezis, H.: Analyse fonctionnelle. Masson 1983
del Pino, M., Felmer, P.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. PDE 4, 121-137 (1996)
del Pino, M., Felmer, P.: Multi-peak bound states of nonlinear Schrödinger equations. Ann. IHP, Analyse Nonlineaire 15, 127-149 (1998)
del Pino, M., Felmer, P.: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324 (1), 1-32 (2002)
del Pino, M., Felmer, P., Tanaka, K.: An elementary construction of complex patterns in nonlinear Schrödinger equations. Nonlinearity 15 (5), 1653-1671 (2002)
Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (3), 397-408 (1986)
Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear equations in \({\bf R}^N\). In: Nachbin, L. (ed.) Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies 7A, pp. 369-402. Academic Press 1981
Grossi, M.: Some results on a class of nonlinear Schrödinger equations. Math. Zeit. 235, 687-705 (2000)
Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Comm. Partial Differential Equations 21, 787-820 (1996)
Jeanjean, L.: On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer-type problem set on \({\bf R}^N\). Proc. Roy. Soc. Edinburgh 129A, 787-809 (1999)
Jeanjean, L., Tanaka, K.: A positive solution for an asymptotically linear elliptic problem on \({\bf R}^N\) autonomous at infinity. ESAIM Control Optim. Calc. Var. 7, 597-614 (2002)
Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \({\bf R}^N\). Proc. Amer. Math. Soc. 131, 2399-2408 (2003)
Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Advances Diff. Eq. 5, 899-928 (2000)
Li, Y.-Y.: On asingularly perturbed elliptic equation. Adv. Differential Equations 2, 955-980 (1997)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109-145, (1984) and II. l Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 223-283 (1984)
Nakashima, K., Tanaka, K.: Clustering layers and boundary layers in spatially inhomogeneous phase transition problems. Ann. I. H. Poincaré Anal. Non Linéaire 20, 107-143 (2003)
Oh, Y.-G.: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V) a . Comm. Partial Differential Equations 13 (12), 1499-1519 (1988)
Oh, Y.-G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131 (2), 223-253 (1990)
Pistoia, A.: Multi-peak solutions for a class of nonlinear Schrödinger equations. NoDEA Nonlinear Diff. Eq. Appl. 9, 69-91 (2002)
Rabinowitz, P.: On a class of nonlinear Schrödinger equations. Z. Angew Math Phys 43, 270-291 (1992)
Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511-517 (1984)
Stuart, C.A.: Personal communication. Summer 2000
Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys. 153, 229-244 (1993)
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Received: 8 November 2003, Accepted: 18 November 2003, Published online: 2 April 2004
Mathematics Subject Classification (2000):
35B25, 35J65, 58E05
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Jeanjean, L., Tanaka, K. Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Cal Var 21, 287–318 (2004). https://doi.org/10.1007/s00526-003-0261-6
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DOI: https://doi.org/10.1007/s00526-003-0261-6