×

The dual variational principle and elliptic problems with discontinuous nonlinearities. (English) Zbl 0687.35033

In this paper the problem \[ (*)\quad \Delta u+f(u)=p(x)\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] \(\Omega\) a bounded domain in \({\mathbb{R}}^ N\), is studied. The nonlinearity f is allowed to have discontinuities and \(p\in L^ 2(\Omega)\). The discontinuities are such that
i) \(f\in C({\mathbb{R}}-A)\) where \(A\subset {\mathbb{R}}\) is a set with no finite accumulation points,
ii) \(h(s):=ms+f(s)\) is strictly increasing for some \(m\geq 0.\)
Solutions of (*) are assumed to be in \(W^ 1_ 0(\Omega)\cap W^{2,2}(\Omega)\) and \[ -\Delta u+p\in \hat f(u)\quad a.e.\quad in\quad \Omega, \] where \[ \hat f(s)=f(s),\quad s\not\in A;\quad \hat f(s)=[f(a- ),f(a+)],\quad u\in A. \] Using variational techniques the authors prove existence theorems in a number of interesting situations.
Reviewer: R.Sperb

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambrosetti, A.; Rabinowitz, P. H., J. Funct. Anal., 14, 349-381 (1973)
[2] Ambrosetti, A.; Srikanth, P. N., J. Math. Phys. Sci., 18, 441-451 (1984)
[3] Brezis, H.; Coron, J. M.; Nirenberg, L., Comm. Pure Appl. Math., 33, 667-689 (1980)
[4] Cerami, G., Rend. Circ. Mat Palermo, 32, 336-357 (1983)
[5] Chang, K. C., J. Math. Anal. Appl., 80, 102-129 (1981)
[6] Clarke, F., J. Differential Equations, 40, 1-6 (1981)
[7] Clarke, F.; Ekeland, I., Comm. Pure Appl. Math., 33, 103-116 (1980)
[8] Prodi, G.; Ambrosetti, A., Analisi Non Lineare (1973), Quaderni della Scuola Normale Superiore: Quaderni della Scuola Normale Superiore Pisa · Zbl 0352.47001
[9] P. H. Rabinowitzin; P. H. Rabinowitzin
[10] Stampacchia, G., Ann. Inst. Fourier (Grenoble), 15, 189-258 (1965)
[11] Stuart, C. A.; Toland, J. F., J. London Math. Soc., 21, 329-335 (1980)
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.