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We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation - Δ p u = λ | u | p - 2 u + a (x) | u | q - 2 u in a bounded domain Ω ⊂ R N , where 1 < q < p , λ ∈ R , and a is a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in { x ∈ Ω : a (x) > 0 } , when the parameter λ lies in a neighborhood of the critical value λ ∗ : = inf ∫ Ω | ∇ u | p d x / ∫ Ω | u | p d x : u ∈ W 0 1 , p (Ω) \ { 0 } , ∫ Ω a | u | q d x ≥ 0 . Among main results, we show that if p > 2 q and either ∫ Ω a φ p q d x = 0 or ∫ Ω a φ p q d x > 0 is sufficiently small, then such solutions do exist in a right neighborhood of λ ∗ . Here φ p is the first eigenfunction of the Dirichlet p-Laplacian in Ω . This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case q > p or in the sublinear case q < p = 2 the nonexistence takes place for any λ ≥ λ ∗ . Moreover, we prove that if p > 2 q and ∫ Ω a φ p q d x > 0 is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of λ ∗ , two of which are strictly positive in { x ∈ Ω : a (x) > 0 } . [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem for the nonhomogeneous equation - Δ p u - Δ q u = α | u | p - 2 u + β | u | q - 2 u + f (x) in a bounded domain, where p ≠ q , and α , β ∈ R are parameters. We explore assumptions on α and β that guarantee the resolvability of the considered problem. Moreover, we introduce several curves on the (α , β) -plane allocating sets of parameters for which the problem has or does not have positive or sign-changing solutions, provided f is of a constant sign. [ABSTRACT FROM AUTHOR]

This paper provides a positive solution for the $(p,q)$ -Laplace equation in $\mathbb{R}^{N}$ with a nonlinear term depending on the gradient. The solution is constructed as the limit of positive solutions in bounded domains. Strengthening the growth condition, it is shown that the solution is also bounded. The positivity of the solution is obtained through a new comparison principle. Finally, under a stronger growth condition, we show the existence of a vanishing at infinity solution. [ABSTRACT FROM AUTHOR]

We study the existence and non-existence of positive solutions for the ( p, q)-Laplace equation $$-\Delta _p u -\Delta _q u=\alpha |u|^{p-2}u+\beta |u|^{q-2}u$$ , where $$p \ne q$$ , under the zero Dirichlet boundary condition in $$\Omega $$ . The main result of our research is the construction of a continuous curve in $$(\alpha ,\beta )$$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $$\Omega $$ for which the corresponding first Dirichlet eigenvalue of $$-\Delta _p$$ is not monotone w.r.t. $$p > 1$$ . [ABSTRACT FROM AUTHOR]

We provide existence results of multiple solutions for quasilinear elliptic equations depending on a parameter under the Neumann and Dirichlet boundary condition. Our main result shows the existence of two opposite constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the solution. The studied equations contain the $$p$$-Laplacian problems as a special case. Our approach is based on variational methods combining super- and sub-solution and the existence of critical points via descending flow. [ABSTRACT FROM AUTHOR]

By variational methods and Morse theory, we prove the existence of uncountably many $$(\alpha ,\beta )\in \mathbb R ^2$$ for which the equation $$-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}$$ in $$\Omega $$ , has a sign changing solution under the Neumann boundary condition, where a map $$A$$ from $$\overline{\Omega }\times \mathbb R ^N$$ to $$\mathbb R ^N$$ satisfying certain regularity conditions. As a special case, the above equation contains the $$p$$ -Laplace equation. However, the operator $$A$$ is not supposed to be $$(p-1)$$ -homogeneous in the second variable. In particular, it is shown that generally the Fučík spectrum of the operator $$-\mathrm{div}\, A(x, \nabla u)$$ on $$W^{1,p}(\Omega )$$ contains some open unbounded subset of $$\mathbb R ^2$$ . [ABSTRACT FROM AUTHOR]

By variational methods, we prove the existence of a non-trivial solution for the quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition. We also provide existence results for positive, negative and non-trivial multiple solutions. The studied equations contain the p-Laplacian problems as a special case. [ABSTRACT FROM AUTHOR]