## Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions.(English)Zbl 1437.35224

Summary: Consider the semilinear Schrödinger equation $\begin{cases} - \triangle u + V ( x ) u = f ( x , u ) , x \in \mathbb{R}^N , \\ u \in H^1 ( \mathbb{R}^N ) , \end{cases}$
where both $$V(x)$$ and $$f(x, u)$$ are periodic in $$x$$, 0 belongs to a spectral gap of $$- \triangle + V$$, and $$f(x, u)$$ is subcritical and allowed to be super-linear at some $$x \in \mathbb{R}^N$$ and asymptotically linear at the other $$x \in \mathbb{R}^N$$. In the existing works in the literature, it is commonly assumed that $$\lim_{| u | \to \infty} \frac{ \int_0^u f ( x , s ) \operatorname{d} s}{ u^2} = \infty$$ uniformly in $$x \in \mathbb{R}^N$$, to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition $$\lim_{| u | \to \infty} \frac{ \int_0^u f ( x , s ) \operatorname{d} s}{ u^2} = \infty$$, a.e. $$x \in G$$ just for some domain $$G \subset \mathbb{R}^N$$.

### MSC:

 35J20 Variational methods for second-order elliptic equations 35J61 Semilinear elliptic equations
Full Text:

### References:

 [1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 [2] Bartsch, T.; Wang, Z.-Q., Existence and multiplicity results for some superlinear elliptic problems on $$\mathbb{R}^N$$, Commun. Partial Differ. Equ., 20, 1725-1741 (1995) · Zbl 0837.35043 [3] Chen, S. T.; Tang, X. H., Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Disc. Contin. Dyn. Syst. A, 38, 2333-2348 (2018) · Zbl 1398.35026 [4] Chen, S. T.; Tang, X. H., On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations (2019) [5] Chen, S. T.; Fiscella, A.; Pucci, P.; Tang, X. H., Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations (2019) [6] Coti Zelati, V.; Rabinowitz, P. H., Homoclinic type solutions for a semilinear elliptic PDE on $$\mathbb{R}^N$$, Commun. Pure Appl. Math., XIV, 1217-1269 (1992) · Zbl 0785.35029 [7] Ding, Y.; Lee, C., Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222, 137-163 (2006) · Zbl 1090.35077 [8] Ding, Y.; Luan, S. X., Multiple solutions for a class of nonlinear Schrödinger equations, J. Differential Equations, 207, 423-457 (2004) · Zbl 1072.35166 [9] Ding, Y.; Szulkin, A., Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29, 3, 397-419 (2007) · Zbl 1119.35082 [10] Edmunds, D. E.; Evans, W. D., Spectral Theory and Differential Operators (1987), Clarendon Press: Clarendon Press Oxford · Zbl 0628.47017 [11] Egorov, Y.; Kondratiev, V., On Spectral Theory of Elliptic Operators (1996), Birkhäuser: Birkhäuser Basel · Zbl 0855.35001 [12] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3, 441-472 (1998) · Zbl 0947.35061 [13] Li, G. B.; Szulkin, A., An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4, 763-776 (2002) · Zbl 1056.35065 [14] Li, G. B.; Wang, C., The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn., Math., 36, 2, 461-480 (2011) · Zbl 1234.35095 [15] Li, Y. Q.; Wang, Z.-Q.; Zeng, J., Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 23, 829-837 (2006) · Zbl 1111.35079 [16] Liu, S., On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45, 1-9 (2012) · Zbl 1247.35149 [17] Liu, Z. L.; Wang, Z.-Q., On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4, 561-572 (2004) [18] Mederski, J., Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Commun. Partial Differ. Equ., 41, 9, 1426-1440 (2016) · Zbl 1353.35267 [19] Pankov, A., Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73, 259-287 (2005) · Zbl 1225.35222 [20] Pavia, F. O.D.; Kryszewski, W.; Szulkin, A., Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term, Proc. Am. Math. Soc., 145, 4783-4794 (2017) · Zbl 1375.35119 [21] Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291 (1992) · Zbl 0763.35087 [22] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0609.58002 [23] Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257, 12, 3802-3822 (2009) · Zbl 1178.35352 [24] Struwe, M., Variational Methods (1996), Springer-Verlag: Springer-Verlag Berlin [25] Tang, X. H., New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud., 14, 361-373 (2014) · Zbl 1305.35036 [26] Tang, X. H., Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98, 104-116 (2015) · Zbl 1314.35030 [27] Tang, X. H., Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58, 715-728 (2015) · Zbl 1321.35055 [28] Tang, X. H.; Chen, S. T., Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 55, 110 (2017) · Zbl 1376.35056 [29] Tang, X. H.; Lin, X. Y.; Yu, J. S., Existence of a bound state solution for quasilinear Schrödinger equations, J. Dyn. Differ. Equ., 31, 369-383 (2019) [30] Troestler, C.; Willem, M., Nontrivial solution of a semilinear Schrödinger equation, Commun. Partial Differ. Equ., 21, 1431-1449 (1996) · Zbl 0864.35036 [31] Willem, M., Minimax Theorems (1996), Birkhäuser: Birkhäuser Boston · Zbl 0856.49001 [32] Willem, M.; Zou, W. M., On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52, 109-132 (2003) · Zbl 1030.35068
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.