An energy constrained method for the existence of layered type solutions of NLS equations

Alessio, Francesca; Montecchiari, Piero

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 4, pp. 725-749.

We study the existence of positive solutions on ${ℝ}^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f\left(u\right)$ where $N⩾1$ and f is modeled on the power case $f\left(u\right)={\left|u\right|}^{p-1}u$. Denoting with c the mountain pass level of $V\left(u\right)=\frac{1}{2}{\parallel u\parallel }_{H^1\left({ℝ}^N\right)}^2-{\int }_{{ℝ}^N}F\left(u\right)dx$, $u\in H^1\left({ℝ}^N\right)$ ($F\left(s\right)={\int }_0^{s}f\left(t\right)dt$), we show, via a new energy constrained variational argument, that for any $b\in [0,c)$ there exists a positive bounded solution $v_b\in C^2\left({ℝ}^{N+1}\right)$ such that $E_{v_b}\left(y\right)=\frac{1}{2}{\parallel {\partial }_y{v}_b(·,y)\parallel }_{L^2\left({ℝ}^N\right)}^2-V\left(v_b(·,y)\right)=-b$ and $v(x,y)\to 0$ as $\left|x\right|\to +\infty$ uniformly with respect to $y\in ℝ$. We also characterize the monotonicity, symmetry and periodicity properties of $v_b$.

[1] F. Alessio, Stationary layered solutions for a system of Allen–Cahn type equations, Indiana Univ. Math. J. (2013) | MR | Zbl

[2] F. Alessio, L. Jeanjean, P. Montecchiari, Stationary layered solutions in ${ℝ}^2$ for a class of non autonomous Allen–Cahn equations, Calc. Var. Partial Differential Equations 11 no. 2 (2000), 177 -202 | MR | Zbl

[3] F. Alessio, P. Montecchiari, Entire solutions in ${ℝ}^2$ for a class of Allen–Cahn equations, ESAIM Control Optim. Calc. Var. 11 (2005), 633 -672 | EuDML | Numdam | MR | Zbl

[4] F. Alessio, P. Montecchiari, Multiplicity of entire solutions for a class of almost periodic Allen–Cahn type equations, Adv. Nonlinear Stud. 5 (2005), 515 -549 | MR | Zbl

[5] F. Alessio, P. Montecchiari, Brake orbits type solutions to some class of semilinear elliptic equations, Calc. Var. Partial Differential Equations 30 no. 51 (2007), 83 | MR | Zbl

[6] A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press (1995) | MR | Zbl

[7] H. Berestycki, T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires, C. R. Acad. Sci. Paris 293 no. 9 (1981), 489 -492 | MR | Zbl

[8] H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313 -345 | MR | Zbl

[9] H. Berestycki, T. Gallouët, O. Kavian, Équations de Champs scalaires Euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math. 297 no. 5 (1983), 307 -310 | MR | Zbl

[10] J. Busca, P. Felmer, Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. 13 (2001), 181 -211 | MR | Zbl

[11] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Mathematicos vol. 26 , IM-UFRJ, Rio de Janeiro (1989)

[12] T. Cazenave, P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 no. 4 (1982), 549 -561 | MR | Zbl

[13] E.N. Dancer, New solutions of equations on ${ℝ}^n$ , Ann. Sc. Norm. Super. Pisa Cl. Sci. 30 (2001), 535 -563 | EuDML | Numdam | MR | Zbl

[14] M. Del Pino, M. Kowalczyk F Pacard, J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math. 224 no. 4–10 (2010), 1462 -1516 | MR | Zbl

[15] C. Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 no. 4 (2008), 904 -933 | MR | Zbl

[16] C. Gui, A. Malchiodi, H. Xu, Axial symmetry of some steady state solutions to nonlinear Schroedinger equations, Proc. Amer. Math. Soc. 139 no. 3 (2011), 1023 -1032 | MR | Zbl

[17] L. Jeanjean, K. Tanaka, A remark on least energy solutions in ${ℝ}^N$ , Proc. Amer. Math. Soc. 131 (2003), 2399 -2408 | MR | Zbl

[18] M.K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p$ in ${ℝ}^n$ , Arch. Ration. Mech. Anal. 105 no. 3 (1989), 243 -266 | MR | Zbl

[19] Y. Li, W. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in ${ℝ}^N$ , Comm. Partial Differential Equations 18 no. 5–6 (1993), 1043 -1054 | MR | Zbl

[20] P.L. Lions, Symétrie et Compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315 -334 | MR | Zbl

[21] A. Malchiodi, Some new entire solutions of semilinear elliptic equations on ${ℝ}^n$ , Adv. Math. 221 no. 6 (2009), 1843 -1909 | MR | Zbl

[22] P. Montecchiari, Multiplicity results for a class of semilinear elliptic equations on ${ℝ}^m$ , Rend. Semin. Mat. Univ. Padova 95 (1996), 1 -36 | EuDML | Numdam | MR

[23] E.H. Lieb, M. Loss, Analysis, Grad. Stud. Math. vol. 14 , American Mathematical Society, Providence, RI (2001) | MR

[24] L. Nirenberg, Lectures on Linear Partial Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 17 , American Mathematical Society (1973) | MR | Zbl

[25] L. Nirenberg, On Elliptic partial differential equations, Ann. Sc. Norm. Pisa 13 (1959), 116 -162 | EuDML | Numdam | MR | Zbl

[26] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 65 , American Mathematical Society, Providence, RI (1986) | MR

[27] J. Serrin, M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 no. 3 (2000), 897 -923 | MR | Zbl

[28] C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse, Appl. Math. Sci. vol. 139 , Springer-Verlag, New York (1999) | MR | Zbl

[29] W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149 -162 | MR | Zbl

[30] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 no. 4 (1983), 567 -576 | MR | Zbl