# The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities

## Abstract

This paper is concerned with the existence of semiregular solutions to the Dirichlet problem for an equation of elliptic type with discontinuous nonlinearity and when the differential operator is not assumed to be formally self-adjoint. Theorems on the existence of semiregular (positive and negative) solutions for the problem under consideration are given, and a principle of upper and lower solutions giving the existence of semiregular solutions is established. For positive values of the spectral parameter, elliptic spectral problems with discontinuous nonlinearities are shown to have nontrivial semiregular (positive and negative) solutions. Bibliography: 32 titles.

- Authors:

- Chelyabinsk State University, Chelyabinsk (Russian Federation)
- Saint Petersburg State University, St. Petersburg (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22590450

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 206; Journal Issue: 9; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; DIRICHLET PROBLEM; EQUATIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS

### Citation Formats

```
Pavlenko, V N, and Potapov, D K.
```*The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities*. United States: N. p., 2015.
Web. doi:10.1070/SM2015V206N09ABEH004496.

```
Pavlenko, V N, & Potapov, D K.
```*The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities*. United States. doi:10.1070/SM2015V206N09ABEH004496.

```
Pavlenko, V N, and Potapov, D K. Wed .
"The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities". United States. doi:10.1070/SM2015V206N09ABEH004496.
```

```
@article{osti_22590450,
```

title = {The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities},

author = {Pavlenko, V N and Potapov, D K},

abstractNote = {This paper is concerned with the existence of semiregular solutions to the Dirichlet problem for an equation of elliptic type with discontinuous nonlinearity and when the differential operator is not assumed to be formally self-adjoint. Theorems on the existence of semiregular (positive and negative) solutions for the problem under consideration are given, and a principle of upper and lower solutions giving the existence of semiregular solutions is established. For positive values of the spectral parameter, elliptic spectral problems with discontinuous nonlinearities are shown to have nontrivial semiregular (positive and negative) solutions. Bibliography: 32 titles.},

doi = {10.1070/SM2015V206N09ABEH004496},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 9,

volume = 206,

place = {United States},

year = {2015},

month = {9}

}

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