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Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the Φ\Phi -Laplacian operator, and the reaction term can be nonmonotone. The main tools employed are the local minimum theorem and the Mountain pass theorem, together with the truncation technique. Global C1,τ{C}^{1,\tau } regularity of solutions is also investigated, chiefly via a priori estimates and perturbation techniques.

A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.

We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the Lq-norm of the solution and the p-Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their Lq-norms.

The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via linking theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualified solutions.

The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a p (x) -Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč's version of the celebrated Díaz–Saá's inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each "positive bump" of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed. [ABSTRACT FROM AUTHOR]

In this paper, we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sub- and supersolutions. Under very general assumptions on the data, we then apply our result to get infinitely many solutions. Moreover, we discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem. [ABSTRACT FROM AUTHOR]

Some recent existence, multiplicity, and uniqueness results for singular $ p $-Laplacian systems either in bounded domains or in the whole space are presented, with a special attention to the case of convective reactions. A extensive bibliography is also provided. [ABSTRACT FROM AUTHOR]

The variational methods are adopted for establishing the existence of at least two nontrivial solutions for a Dirichlet problem driven by a non-homogeneous di¤erential operator of p-Laplacian type. A large class of nonlinear terms is considered, covering the concave-convex case. In particular, two positive solutions to the problem are obtained under a (p - 1)-superlinear growth at infinity, provided that a behaviour less than (p - 1)-linear of the nonlinear term in a suitable set is requested. [ABSTRACT FROM AUTHOR]

The paper sets forth a new type of variational problem without any ellipticity or monotonicity condition. A prototype is a differential inclusion whose driving operator is the competing weighted (p, q)-Laplacian Dpu + mDqu with m 2 R. Local and nonlocal boundary value problems fitting into this nonstandard setting are examined. [ABSTRACT FROM AUTHOR]

We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet p-Laplacian. We consider three cases. In the first the perturbation is (p-1)-sublinear near +∞, while in the second the perturbation is (p - 1)-superlinear near +∞ and in the third we do not require asymptotic condition at +∞. Using variational methods together with truncation and comparison techniques, we show that for λ ∈ (0;λ1) - λ > 0 is the parameter and λ1 being the principal eigenvalue of (-Δp;W01,p(Ω)) - we have positive solutions, while for λ ≥ λ1, no positive solutions exist. In the "sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at +∞, provided that the perturbation is damped by a parameter. [ABSTRACT FROM AUTHOR]