We consider a nonlinear eigenvalue problem driven by the double phase differential operator. We prove two existence theorems, both producing a continuous spectrum and the first generates eigenfunctions which blow up in the W-0(1,theta) (Omega) boolean AND L (R) (Omega)-norm ( 1 < q < p < r < q & lowast; ) , while the second generates eigenfunctions which vanish in the W-0(1,theta)(Omega) boolean AND L-infinity (Omega) -norm as lambda -> 0(+).

We consider a perturbed eigenvalue problem for the unbalanced (p, q)-Laplacian. The perturbation exhibits the combined effects of concave and convex nonlinearities. Using the Nehari method, we show that for all small values of the parameter lambda > 0 the problem has at least two ground state (least energy) solutions, one with positive energy and the second with negative energy. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We consider a Dirichlet problem driven by a differential operator with unbalanced growth and a reaction exhibiting the combined effects of a parametric singular term and a resonant perturbation. Using a combination of variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least two bounded positive solutions.

We consider a degenerate p-Laplacian equation, with a reaction that exhibits the competing effects of a parametric concave term and of a sign-changing convex perturbation. Using variational tools and critical groups, we prove an existence and multiplicity result which is global in the parameter. In the process, we also prove some general results of independent interest.

We consider a nonlinear, nonhomogeneous singular elliptic problem with [Formula: see text]-growth (unbalanced growth). Using a combination of variational tools with critical groups, we establish the existence of at least two bounded solutions.

We consider a Dirichlet problem driven by the nonautonomous, degenerate p-Laplacian, with a reaction resonant at +/-infinity and at 0. Using variational tools and critical groups (Morse theory), we show that the problem has at least two nontrivial bounded solutions. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study a nonlinear eigenvalue problem driven by a general nonhomogeneous differential operator, involving a reaction term that is singular at x = 0 and becomes superlinear as x → +∞. Unlike the usual case in the literature, the singular term and the perturbation are not decoupled. By using variational methods in combination with truncation and comparison techniques, we establish a global existence and multiplicity theorem with respect to the parameter (eigenvalue) λ > 0. Additionally, we demonstrate the existence of a minimal positive solution u*λ and investigate the continuity and monotonicity properties of the map λ → u*λ.

We study a nonlinear Dirichlet problem eigenvalue driven by a differential operator with unbalanced growth (double phase problem) and a reaction that has the competing effects of a singular term and of a superlinear perturbation. We prove an existence and multiplicity theorem which is global in the parameter lambda > 0.

We consider a Dirichlet problem driven by a double phase differential operator and a reaction which exhibits the combined effects of a parametric singular term and of an indefinite superlinear perturbation. The superlinearity condition on the perturbation is very general. Using variational tools, truncation and comparison techniques and critical groups, we prove an existence and multiplicity result which is global in the parameter (bifurcation-type result).

We consider a nonlinear Dirichlet problem driven by a nonautonomous (p, q)-differential operator and with a reaction having the competing effects of a parametric singular term and a (p - 1)linear perturbation which can be resonant as x -> infinity with respect to the principal eigenvalue of the relevant operator. If the resonance is from the left, then we demonstrate that the problem has a positive solution for all values of the parameter and if the driving differential operator is only the nonautonomous pLaplacian, then the positive solution is unique. On the other hand, if the resonance is from the right, then we prove an existence and multiplicity theorem which is global with respect to the parameter (a bifurcation-type theorem). Also, we conduct a detailed study of the continuity properties of solution multifunction.

We consider a singular Dirichlet problem driven by a variable unbalanced growth differential operator and a singular reaction with an unbounded coefficient and an exponent η∈ C(Ω) which is less than 1 only near ∂Ω . Using approximations, we show that the problem has a unique bounded solution.

We consider an elliptic equation driven by the nonautonomous degenerate p-Laplacian and with a reaction which exhibits a parametric singular term and a superlinear perturbation. Using the theory of weighted spaces together with variational tools, truncations, comparisons techniques, and critical groups, we prove on existence and multiplicity theorem, which is global in the parameter (i.e., a bifurcation-type theorem).

We consider a nonlinear Dirichlet problem with gradient dependence. The features of this paper are twofold: (i) the problem is driven by a general nonlinear nonhomogeneous differential operator with Uhlenbeck-Lieberman structure; (ii) the reaction blows-up at the origin and it is gradient dependent. Using a topological approach based on fixed point theory, we show that for all small values of lambda > 0 there are "eigenvalues" of the problem with smooth corresponding eigenfunctions.

We prove the existence and multiplicity of solutions for a variable-exponent double-phase problem with parametric logistic reaction term. Combining variational and truncation methods with homological critical group theory, we prove the existence of at least one or two solutions with respect to a positive parameter, i.e., a bifurcation result.

We examine three singular Dirichlet problems driven by the double phase operator. One of the problems is nonparametric and the other two are parametric. In all problems, the perturbation is “superlinear”, but does not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity results for the problems. For the parametric problems, the results are global in the parameter λ >0 . Our approach uses variational tools from the critical point theory, truncations and comparisons and critical groups.

We consider a Dirichlet problem driven by a nonlinear nonhomogeneous differential operator, with a reaction which exhibits the combined effects of a parametric singular term and of a Carathéodory gradient dependent perturbation. Using the theory of nonlinear operators of monotone type, we show that for all small values of the parameter the problem has at least one positive smooth solution.

We consider a nonlinear Robin problem driven by a differential operator with unbalanced growth and a reaction which exhibits the competing effects of a parametric concave (sublinear) term and of a convex (superlinear) term. Using the Nehari method, we show that for all small values of the parameter, the problem has two bounded, ground state (least energy) solutions. In the process of the proof, we establish some auxiliary results which are of independent interest.

We consider a nonlinear Dirichlet equation driven by the weighted p-Laplacian with weight a(· ) which is not bounded away from zero (degenerate problem). The reaction (right hand side) is parametric and exhibits the combined effects of a singular term and of indefinite (sign-changing), ( p-1 )-superlinear perturbation. Using variational tools, truncations, comparisons and critical groups, we show that for all small values of the parameter, the problem has at least two bounded positive solutions.

In this work, we investigate the existence of at least three solutions to a variable exponent double-phase problem with a reaction term that is only locally Lipschitz continuous. Additionally, we analyse the sign properties of these solutions. Specifically, we establish the existence of two constant-sign solutions, one positive and one negative, using the Mountain Pass Theorem. The third solution, which is nodal (i.e., it changes sign), is obtained via the Nehari manifold approach. Finally, we demonstrate that the nodal solution has exactly two nodal domains.

We consider a nonlinear eigenvalue problem driven by the anisotropic (p, q)-Laplacian. Using variational tools, truncations, comparisons and critical groups, we show that for all small values of the parameter, the problem has extremal constant sign solutions and nodal solutions. These solutions are ordered and vanish in C_0^1(Ω) as λ→ 0^+ .