Abstract
In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.
Highlights
IntroductionW 1,p(z) (Ω) presents difficulties about the density of smooth functions (Meyers-Serrin [13]), the Sobolev inequality, and embedding theorems (Edmunds-Rákosník [14], Kováčik-Rákosník [12])
W 1,p(z) (Ω) presents difficulties about the density of smooth functions (Meyers-Serrin [13]), the Sobolev inequality, and embedding theorems (Edmunds-Rákosník [14], Kováčik-Rákosník [12]). This means that the passage from the constant exponent setting to the variable exponent setting needs attention to special cases, and some challenging open problems remain
Sufficient criteria of the existence of weak solutions to local Dirichlet (r (u), s(u))-problems with certain nonlinearities have been presented in this work
Summary
W 1,p(z) (Ω) presents difficulties about the density of smooth functions (Meyers-Serrin [13]), the Sobolev inequality, and embedding theorems (Edmunds-Rákosník [14], Kováčik-Rákosník [12]) This means that the passage from the constant exponent setting to the variable exponent setting needs attention to special cases, and some challenging open problems remain (for further details, we refer to Barile-Figueiredo [15] and Cencelj-Rădulescu-Repovš [16], and the references therein). There are two key papers dealing with the single r (u)-Laplacian equation, both with homogeneous Dirichlet boundary conditions (see Andreianov-Bendahmane-Ouaro [17] and Chipot-de Oliveira [18]) In these papers, the authors established existence and uniqueness results, assuming the nonlinearity N has an appropriate structure.
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