zbMATH — the first resource for mathematics

Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions. (English) Zbl 1034.35041
The authors study the problem \[ \begin{aligned} -\Delta u &= \lambda u^q + u^r \quad\text{in } \Omega, \cr u &> 0 \quad\text{in } \Omega, \cr B(u) &= 0 \quad\text{on } \partial\Omega, \end{aligned} \eqno(*) \] where \(\Omega\subset{\mathbb R}^N\), \(N\geq 3\), is a smooth bounded domain, \(1<r<2^*-1=(N+2)/(N-2)\), \(0<q<r\), and \(B\) is a mixed boundary operator of the form \[ B(u)= u \chi_{\Sigma_1} + {{\partial u}\over{\partial \nu}}\chi_{\Sigma_2}, \] where \(\Sigma_1\) and \(\Sigma_2\) are smooth \((N-1)\)-dimensional submanifolds of \(\partial\Omega\) such that \(\Sigma_1\cap\Sigma_2=\emptyset\), \(\overline\Sigma_1\cup\overline\Sigma_2=\partial\Omega\) and \(\overline\Sigma_1\cup\overline\Sigma_2=\Gamma\) is a smooth \((N-2)\)-dimensional submanifold of \(\partial\Omega\).
The main results are existence, nonexistence and multiplicity results, and a priori \(L^\infty\) estimates for solutions of \((*)\). These are proved by a blow-up argument that relies on some refinements of the global Hölder estimates for solutions of mixed boundary value problems established by G. Stampacchia [Ann. Mat. Pura Appl. (4) 51, 1–37 (1960; Zbl 0204.42001)].

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
Full Text: DOI
[1] Ambrosetti, A.; Brezis, H.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. funct. anal., 122, 2, 519-543, (1994) · Zbl 0805.35028
[2] Ambrosetti, A.; Garcı́a Azorero, J.; Peral Alonso, I., Quasilinear equations with a multiple bifurcation, Differential integral equations, 10, 1, 37-50, (1997) · Zbl 0879.35021
[3] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[4] Berestycki, H.; Grossi, M.; Pacella, F., A nonexistence theorem for an equation with critical Sobolev exponent in the half space, Manuscripta math., 77, 2-3, 265-281, (1992), (English summary) · Zbl 0796.35050
[5] Berestycki, H.; Nirenberg, L., On the method of moving planes and the sliding method, Bol. soc. brasil. mat. (NS), 22, 1, 1-37, (1991) · Zbl 0784.35025
[6] Berestycki, H.; Nirenberg, L.; Varadhan, S.R.S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. pure appl. math., 47, 1, 47-92, (1994) · Zbl 0806.35129
[7] Brezis, H.; Nirenberg, L., H1 versus \(C\^{}\{1\}\) local minimizers, C. R. acad. sci. Paris Sér. I math., 317, 5, 465-472, (1993) · Zbl 0803.35029
[8] Chen, W.X.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke math. J., 63, 3, 615-622, (1991) · Zbl 0768.35025
[9] L. Damascelli, F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, preprint. · Zbl 1330.35146
[10] Dávila, J., A strong maximum principle for the Laplace equation with mixed boundary condition, J. funct. anal., 183, 231-244, (2001) · Zbl 0979.35037
[11] Garcı́a Azorero, J.; Peral Alonso, I., Multiplicity of solutions for elliptic problems with critical exponent or with non-symmetric term, Trans. amer. math. soc., 323, 2, 877-895, (1991) · Zbl 0729.35051
[12] Ghoussoub, N.; Preiss, D., A general mountain pass principle for locating and classifying critical points, Ann. inst. H. Poincaré anal. non linéare, 6, 321-330, (1989) · Zbl 0711.58008
[13] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. partial differential equations, 6, 8, 883-901, (1981) · Zbl 0462.35041
[14] Lin, C.S.; Ni, W.M.; Takagi, I., Large amplitude stationary solutions to a chemotaxis system, J. differential equations, 72, 1, 1-27, (1988) · Zbl 0676.35030
[15] Malý, J.; Ziemer, W., Fine regularity of solutions of elliptic partial differential equations, Mathematical surveys and monograph, Vol. 51, (1997), American Mathematical Society Providence, RI · Zbl 0882.35001
[16] Miranda, C., Sul problema misto per le equazioni lineari ellittiche, Ann. mat. pura appl., 4, 39, 279-303, (1955) · Zbl 0066.34301
[17] Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[18] Shamir, E., Regularization of second-order elliptic problems, Israel J. math., 6, 150-168, (1968) · Zbl 0157.18202
[19] Stampacchia, G., Problemi al contorno ellitici, con dati discontinui, dotati di soluzioni hölderiane, Ann. mat. pura appl., 51, 4, 1-37, (1960), (Italian) · Zbl 0204.42001
[20] Terracini, S., Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential integral equations, 8, 8, 1911-1922, (1995) · Zbl 0835.35055
[21] Terracini, S., On positive solutions to a class of equations with a singular coefficient and critical exponent, Adv. differential equation, 1, 2, 241-264, (1996) · Zbl 0847.35045
[22] Whyburn, G.T., Topological analysis, (1964), Princeton University Press Princeton, NJ
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.