×

zbMATH — the first resource for mathematics

Concentration of solutions for some singularly perturbed mixed problems: asymptotics of minimal energy solutions. (English) Zbl 1194.35037
This paper deals with positive solutions to the problem
\[ -\varepsilon^2\Delta u+u=u^p \text{ in } \Omega, \quad 1<p<\frac{n+2}{n-2} \tag{1} \]
together with the boundary conditions
\[ \dfrac{\partial u}{\partial \nu}=0 \text{ on } \partial_{\mathcal N}\Omega ;\quad u=0 \text{ on }\partial_{\mathcal D} \Omega \tag{2} \] where \(\varepsilon\) is a small positive parameter, \(\Omega\) is a smooth bounded subset of \(\mathbb R^n\), \(\partial_{\mathcal N}\Omega\) and \(\partial _{\mathcal D}\Omega\) are two disjoint open subsets of the boundary \(\partial\Omega\) such that \(\partial\Omega=\overline{\partial_{\mathcal N}\Omega} \cup\overline{\partial_{\mathcal D}\Omega}\) and \(\mathcal I_{\Omega}=:\overline{\partial_{\mathcal N}\Omega} \cap\overline{\partial_{\mathcal D}\Omega}\) is an imbedded hypersurface. Let \(H\) denote the mean curvature of \(\partial\Omega.\) Then, the main result states that, as \(\varepsilon\) tends to zero, the least energy solution to (1)-(2) has a unique maximum point which converges to \(Q\in\overline {\partial_{\mathcal N}\Omega}\) such that \(H(Q)=\underset\overline{{\partial_{\mathcal N}\Omega}}{\max}H.\) The proof is based on arguments of W. M. Ni and I.Takagi, [Commun. Pure Appl. Math. 44, No. 7, 819–851 (1991; Zbl 0754.35042)], and the authors’ paper [Arch. Ration. Mech. Anal. 196, No. 3, 907–950 (2010; Zbl 1214.35024)].

MSC:
35B25 Singular perturbations in context of PDEs
35B34 Resonance in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ambrosetti, A.; Malchiodi, A., Perturbation methods and semilinear elliptic problems on \(\mathbb{R}^n\), Progr. math., vol. 240, (2005), Birkhäuser
[2] Ambrosetti, A.; Malchiodi, A., Nonlinear analysis and semilinear elliptic problems, Stud. adv. math., vol. 104, (2007), Cambridge Univ. Press Cambridge · Zbl 1125.47052
[3] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[4] Berestycki, H.; Caffarelli, L.; Nirenberg, L., Further qualitative properties for elliptic equations in unbounded domains, Ann. sc. norm. super. Pisa cl. sci. (4), 25, 1-2, 69-94, (1997) · Zbl 1079.35513
[5] Colorado, E.; Peral, I., Eigenvalues and bifurcation for elliptic equations with mixed dirichlet – neumann boundary conditions related to caffarelli – kohn – nirenberg inequalities, Topol. methods nonlinear anal., 23, 2, 239-273, (2004) · Zbl 1075.35014
[6] Damascelli, L.; Gladiali, F., Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. mat. iberoamericana, 20, 1, 67-86, (2004) · Zbl 1330.35146
[7] Dancer, E.N.; Yan, S., Interior and boundary peak solutions for a mixed boundary value problem, Indiana univ. math. J., 48, 4, 1177-1212, (1999) · Zbl 0948.35055
[8] Del Pino, M.; Felmer, P., Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana univ. math. J., 48, 3, 883-898, (1999) · Zbl 0932.35080
[9] J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press · Zbl 1214.35024
[10] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbb{R}^n\), Adv. math. (suppl. stud. A), 7, 369-402, (1981)
[11] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag Berlin/Heidelberg/New York/Tokyo · Zbl 0691.35001
[12] Kwong, M.K., Uniqueness of positive solutions of \(- \operatorname{\Delta} u + u - u^p = 0\) in \(\mathbb{R}^n\), Arch. ration. mech. anal., 105, 243-266, (1989) · Zbl 0676.35032
[13] Lin, C.S.; Ni, W.M.; Takagi, I., Large amplitude stationary solutions to a chemotaxis systems, J. differential equations, 72, 1-27, (1988) · Zbl 0676.35030
[14] Ni, W.M.; Wei, J., On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. pure appl. math., 48, 731-768, (1995) · Zbl 0838.35009
[15] Ni, W.M.; Takagi, I., On the shape of least-energy solution to a semilinear Neumann problem, Comm. pure appl. math., 41, 819-851, (1991) · Zbl 0754.35042
[16] Ni, W.M.; Takagi, I., Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke math. J., 70, 247-281, (1993) · Zbl 0796.35056
[17] Stampacchia, G., Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hölderiane, Ann. mat. pura appl. (4), 51, 1-37, (1960) · Zbl 0204.42001
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.