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Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki–Lions condition

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Abstract

Let \(\varOmega \) be a smooth bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). We consider the following singularly perturbed nonlinear elliptic problem on \(\varOmega \),

$$\begin{aligned} \varepsilon ^2\varDelta v-v+f(v)=0,\quad v>0\ \text {on}\ \varOmega ,\qquad \frac{\partial v}{\partial \nu }=0\quad \text {on}\ \partial \varOmega , \end{aligned}$$

where \(\nu \) is an exterior unit normal vector to \(\partial \varOmega \) and a nonlinearity f satisfies subcritical growth condition. It has been known that for any \(l_0, l_1 \in \mathbb {N} \cup \{ 0 \}\), \(l_0+l_1>0\), there exists a solution \(v_\varepsilon \) of the above problem which exhibits \(l_0\)-boundary peaks and \(l_1\)-interior peaks for small \(\varepsilon >0\) under rather strong conditions on f, such as the linearized non-degeneracy condition for a limiting problem. In this paper, we extend the previous result to more general class of f satisfying Berestycki–Lions conditions which we believe to be almost optimal.

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References

  1. Adams, R.A.: Sobolev spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)

  2. Adimurthi, M.G., Yadava, S.L.: The role of the mean curvature in semilinear Neumann problem involving critical exponent. Commun. Partial Differ. Equ. 20, 591–631 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adimurthi, P.F., Yadava, S.L.: Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity. J. Funct. Anal. 113, 318–350 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Adimurthi, P.F., Yadava, S.L.: Characterization of concentration points and \(L^\infty \)-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent. Differ. Integral Equ. 8, 41–68 (1995)

    MathSciNet  MATH  Google Scholar 

  5. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ao, W., Wei, J., Zeng, J.: An optimal bound on the number of interior spike solutions for the Lin–Ni–Takagi problem. J. Funct. Anal. 265(7), 1324–1356 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations I. Arch. Ration. Mech. Anal. 82, 313–345 (1983)

    MATH  Google Scholar 

  8. Byeon, J.: Singularly perturbed nonlinear Neumann problems with a general nonlinearity. J. Differ. Equ. 244, 2473–2497 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byeon, J., Jeanjean, L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185, 185–200 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Byeon, J., Lee, Y.: Singularly perturbed nonlinear Neumann problems under the conditions of Berestycki and Lions. J. Differ. Equ. 252, 3848–3872 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Byeon, J., Park, J.: Singularly perturbed nonlinear elliptic problems on manifolds. Calc. Var. Partial Differ. Equ. 24, 459–477 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Byeon, J., Tanaka, K.: Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. 15, 1859–1899 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Byeon, J., Tanaka, K.: Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations. Memoirs of the American Mathematical Society, vol. 229, no. 1076. American Mathematical Society. ISBN: 978-0-8218-9163-6 (2014)

  14. Byeon, J., Tanaka, K.: Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains. Cal. Var. Partial Differ. Equ. 50(1–2), 365–397 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cerami, G., Passaseo, D., Solimini, S.: Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients. Commun. Pure Appl. Math. 66(3), 372–413 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Coti-Zelati, V., Rabinowitz, P.: Homoclinic type solutions for a semilinear elliptic PDE on \(\mathbb{R}^n\). Commun. Pure Appl. Math. 45(10), 1217–1269 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. del Pino, M., Felmer, P.L.: Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting. Indiana Univ. Math. J. 48, 883–898 (1999)

    MathSciNet  MATH  Google Scholar 

  18. del Pino, M., Felmer, P.L.: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324, 1–32 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. del Pino, M., Felmer, P.L., Wei, J.: On the role of mean curvature in some singularly perturbed Neumann problems. SIAM J. Math. Anal. 31, 63–79 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. del Pino, M., Wei, J., Yao, W.: Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials. Online published in Calc. Var. Partial Differ. Equ. doi:10.1007/s00526-014-0756-3

  21. Dancer, E.N., Yan, S.: Multipeak solutions for a singularly perturbed Neumann problem. Pac. J. Math. 189, 241–262 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, vol. 224, 2nd edn. Springer, Berlin (1983)

  24. Gui, C.: Multipeak solutions for a semilinear Neumann problem. Duke Math. J. 84, 739–769 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gui, C., Wei, J.: Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Differ. Equ. 158, 1–27 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gui, C., Wei, J.: On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Can. J. Math. 52, 522–538 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gui, C., Wei, J., Winter, M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 47–82 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Han, Q., Lin, F.: Elliptic Partial Differential Equations. Courant Lecture Notes. American Mathematical Society, Providence (2000)

  29. Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \(\mathbb{R}^N\). Proc. Am. Math. Soc. 131, 2399–2408 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jeanjean, L., Tanaka, K.: A note on a mountain pass characterization of least energy solutions. Adv. Nonlinear Stud. 3, 445–455 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, Y.Y.: On a singularly perturbed equation with Neumann boundary condition. Commun. Partial Differ. Equ. 23, 487–545 (1998)

    MathSciNet  MATH  Google Scholar 

  32. Lin, C.S., Ni, W.M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. Differ. Equ. 72, 1–27 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Part II. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1, 223–283 (1984)

    MathSciNet  MATH  Google Scholar 

  34. Ni, W.M., Pan, X., Takagi, I.: Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. Duke Math. J. 67, 1–20 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ni, W.M., Takagi, I.: Point condensation generated by a reaction–diffusion system in axially symmetric domains. Jpn. J. Ind. Appl. Math. 12, 327–365 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ni, W.M., Takagi, I.: On the shape of least-energy solutions to a semilinear Neumann problem. Commun. Pure Appl. Math. 44, 819–851 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ni, W.M., Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247–281 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, Z.Q.: On the existence of multiple, single-peaked solutions for a semilinear Neumann problem. Arch. Ration. Mech. Anal. 120, 375–399 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wei, J.: On the interior spike layer solutions to a singularly perturbed Neumann problem. Tohoku Math. J. 2(50), 159–178 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

J. Seok was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2014R1A1A2054805), and also was supported by the POSCO TJ Park Science Fellowship. The authors would like to express their sincere gratitude to Prof. Byeon and Prof. Tanaka for their direction and encouragement.

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Authors and Affiliations

  1. Center for Advanced Study in Theoretical Sciences, National Taiwan University, No.1, Sec. 4, Roosevelt Road, Taipei, 106, Taiwan

    Youngae Lee

  2. Department of Mathematics, Kyonggi University, 154-42 Gwanggyosan-ro, Yeongtong-gu, Suwon, 443-760, Republic of Korea

    Jinmyoung Seok

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  1. Youngae Lee
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Correspondence to Jinmyoung Seok.

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Lee, Y., Seok, J. Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki–Lions condition. Math. Ann. 367, 881–928 (2017). https://doi.org/10.1007/s00208-016-1412-3

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  • DOI: https://doi.org/10.1007/s00208-016-1412-3

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