×

On the existence of solutions for the critical fractional Laplacian equation in \(\mathbb{R}^N\). (English) Zbl 1470.35411

Summary: We study existence of solutions for the fractional Laplacian equation \(\left(- \Delta\right)^s u + V \left(x\right) u = \left|u\right|^{2^* \left(s\right) - 2} u + f \left(x, u\right)\) in \(\mathbb{R}^N\), \(u \in H^s(R^N)\), with critical exponent \(2^* \left(s\right) = 2 N /(N - 2 s)\), \(N > 2 s\), \(s \in \left(0, 1\right)\), where \(V \left(x\right) \geq 0\) has a potential well and \(f : \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}\) is a lower order perturbation of the critical power \(\left|u\right|^{2^* \left(s\right) - 2} u\). By employing the variational method, we prove the existence of nontrivial solutions for the equation.

MSC:

35R11 Fractional partial differential equations
35B33 Critical exponents in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brézis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communications on Pure and Applied Mathematics, 36, 4, 437-477 (1983) · Zbl 0541.35029
[2] Clapp, M.; Ding, Y., Positive solutions of a Schrödinger equation with critical nonlinearity, Zeitschrift fur Angewandte Mathematik und Physik, 55, 4, 592-605 (2004) · Zbl 1060.35130
[3] Rabinowitz, P. H., On a class of nonliear Schrödinger equations, Zeitschrift für Angewandte Mathematik und Physik, 43, 2, 270-291 (1992) · Zbl 0763.35087
[4] Yang, M. B.; Ding, Y. H., Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part, Communications on Pure and Applied Analysis, 12, 2, 771-783 (2013) · Zbl 1270.35218
[5] Yang, M. B., Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities, Nonlinear Analysis: Theory, Methods & Applications, 75, 13, 5362-5373 (2012) · Zbl 1258.35077
[6] Brézis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communications on Pure and Applied Analysis, 36, 4, 437-477 (1983) · Zbl 0541.35029
[7] Strauss, W. A., Existence of solitary waves in higher dimensions, Communications in Mathematical Physics, 55, 2, 149-162 (1977) · Zbl 0356.35028
[8] Chabrowski, J.; Yang, J., Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Zeitschrift fur Angewandte Mathematik und Physik, 49, 2, 276-293 (1998) · Zbl 0903.35021
[9] Benci, V.; Cerami, G., Existence of positive solutions of the equation \(- ∆ u + a \left(x\right) u = \left``|u\right``|^{\left(N + 2\right) / \left(N - 2\right)}\) in \(\mathbb{R}^N\), Journal of Functional Analysis, 88, 1, 90-117 (1990) · Zbl 0705.35042
[10] Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, Journal of Functional Analysis, 69, 3, 397-408 (1986) · Zbl 0613.35076
[11] Servadei, R.; Valdinoci, E., The Brezis-Nirenberg result for the fractional Laplacian · Zbl 1323.35202
[12] Servadei, R.; Valdinoci, E., Fractional Laplacian equations with critical sobolev exponent · Zbl 1338.35481
[13] Servadei, R., The Yamabe equation in a non-local setting, Advances in Nonlinear Analysis, 2, 3, 235-270 (2013) · Zbl 1273.49011
[14] Servadei, R.; Brézis-Nirenberg, A., A Brezis-Nirenberg result for non-local critical equations in low dimension, Communications on Pure and Applied Analysis, 12, 6, 2445-2464 (2013) · Zbl 1302.35413
[15] Felmer, P.; Quaas, A.; Tan, J. G., Positive solutions of nonlinear Schrödinger equation with fractional Laplacian, Proceedings of the Royal Society of Edinburgh, 142, 6, 1237-1262 (2012) · Zbl 1290.35308
[16] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bulletin des Sciences Mathematiques, 136, 5, 521-573 (2012) · Zbl 1252.46023
[17] Servadei, R.; Valdinoci, E., Mountain Pass solutions for non-local elliptic operators, Journal of Mathematical Analysis and Applications, 389, 2, 887-898 (2012) · Zbl 1234.35291
[18] Palatucci, S.; Dipierro, G.; Valdinoci, E., Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68, 1, 201-216 (2013) · Zbl 1287.35023
[19] Costa, D. G., On a class of elliptic systems in \(\mathbb{R}^N\), Electronic Journal of Differential Equations, 1994, 7, 1-14 (1994)
[20] Chang, K. C., Methods in Nonlinear Analysis (2005), Berlin, Germany: Springer, Berlin, Germany
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.