1,721,240 research outputs found
Variable exponent p(x)-Kirchhoff type problem with convection
Full text linkWe study a nonlinear p(x)-Kirchhoff type problem with Dirichlet boundary condition, in the case of a reaction term depending also on the gradient (convection). Using a topological approach based on the Galerkin method, we discuss the existence of two notions of solutions: strong generalized solution and weak solution. Strengthening the bound on the Kirchhoff type term (positivity condition), we establish existence of weak solution, this time using the theory of operators of monotone type
Semilinear Robin problems driven by the Laplacian plus an indefinite potential
Full text linkWe study a semilinear Robin problem driven by the Laplacian plus an indefinite potential. We consider the case where the reaction term f is a Carathéodory function exhibiting linear growth near ±∞. So, we establish the existence of at least two solutions, by using the Lyapunov-Schmidt reduction method together with variational tools
Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential
Full text linkWe consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation f(z, ·) is (p- 1) -sublinear and then the case where it is (p- 1) -superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter λ∈ R which we specify exactly in terms of principal eigenvalue of the differential operator
A model of capillary phenomena in RN with sub-critical growth
Full text linkThis paper deals with the nonlinear Dirichlet problem of capillary phenomena involving an equation driven by the p-Laplacian-like di¤erential operator in RN. We prove the existence of at least one nontrivial nonnegative weak solution, when the reaction term satisfies a sub-critical growth condition and the potential term has certain regularities. We apply the energy functional method and weaker compactness conditions
Parametric and nonparametric A-Laplace problems: Existence of solutions and asymptotic analysis
Full text linkWe give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A-Laplace operator in a bounded domain ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a suitable parameter goes to 0 + . In this case, we point out the existence of a blow-up phenomenon. The analysis developed in this paper extends and complements various qualitative and asymptotic properties for some cases described by homogeneous differential operators
Pairs of nontrivial smooth solutions for nonlinear Neumann problems
Full text linkWe consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator with a reaction term that exhibits strong resonance at infinity. Using variational tools based on the critical point theory, we prove the existence of two nontrivial smooth solutions
An elliptic equation on n-dimensional manifolds
Full text linkWe consider an elliptic equation driven by a p-Laplacian-like operator, on an n-dimensional Riemannian manifold. The growth condition on the right-hand side of the equation depends on the geometry of the manifold. We produce a nontrivial solution by using a Palais–Smale compactness condition and a mountain pass geometry
The Existence of Solutions for Local Dirichlet (r(u), s(u))-Problems
Full text linkIn 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
A fixed-point problem with mixed-type contractive condition
Full text linkWe consider a fixed-point problem for mappings involving a mixed-type contractive condition in the setting of metric spaces. Precisely, we establish the existence and uniqueness of fixed point using the recent notions of F-contraction and (H;Phi)-contraction
Regularity and Dirichlet Problem for Double-Phase Energy Functionals of Different Power Growth
No full textWe work with a double-phase energy functional exhibiting logL-perturbed p&q-growth. We look for regularity properties of such functional in the setting of Musielak-Orlicz-Sobolev space, by imposing suitable conditions on the data. We further obtain the existence and uniqueness results for the solution of perturbed Dirichlet double-phase problems. We deal with both the cases of uncontrolled growth and controlled growth
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