Positive solutions of nonlinear elliptic equations involving supercritical Sobolev exponents without Ambrosetti and Rabinowitz condition
Autor: | Jéssyca L. F. Melo Gurjão, Luiz F. O. Faria, Anderson L. A. de Araujo |
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Rok vydání: | 2020 |
Předmět: |
Pure mathematics
Sublinear function Applied Mathematics 010102 general mathematics Mathematics::Analysis of PDEs 01 natural sciences Dirichlet distribution 010101 applied mathematics Sobolev space symbols.namesake Compact space symbols Exponent Boundary value problem Ball (mathematics) 0101 mathematics Lp space Analysis Mathematics |
Zdroj: | Calculus of Variations and Partial Differential Equations. 59 |
ISSN: | 1432-0835 0944-2669 |
Popis: | The purpose of the present paper is to study a class of semilinear elliptic Dirichlet boundary value problems in the ball, where the nonlinearities involve the sum of a sublinear variable exponent and a superlinear (may be supercritical) variable exponents of the form $$0\le f(r,u)\le a_1|u|^{p(r)-1}$$ , if $$u\ge 0$$ , where $$r = |x|$$ , $$p(r ) = 2^* + r^{\alpha }$$ , with $$\alpha > 0$$ , and $$2^* = 2N/(N - 2)$$ is the critical Sobolev embedding exponent. We do not impose the Ambrosetti–Rabinowitz condition on the nonlinearity (or some additional conditions) to obtain Palais–Smale or Cerami compactness condition. We employ techniques based on the Galerkin approximations scheme, combining with a Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces (due to do O et al. in Calc Var Partial Differ Equ 55:83, 2016), to establish the existence result. |
Databáze: | OpenAIRE |
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