We deal with some Dirichlet problems involving a nonlocal term. The existence of two nonzero, nonnegative solutions is achieved by applying a recent result by Ricceri.

Two nontrivial solutions are obtained for nonhomogeneous semilinear Schrödinger equations.

We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in ${ℝ}^N$ with boundary of class C², $\alpha \in L^{\infty }\left(\Omega \right)$ with $essin{f}_{\Omega }\alpha >0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $su{p}_{\left|s\right|\le t}\left|g(\cdot ,s)\right|\in L^p\left(\Omega \right)$ and $g(\cdot ,t)\in L^{\infty }\left(\Omega \right)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2\xi ²-{\int }_0^{\xi }f\left(t\right)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough, the above...

We show that, under appropriate structure conditions, the quasilinear Dirichlet problem $$\left\{\begin{array}{ccc}-div\left(\right|\nabla u{|}^{p-2}\nabla u)=f(x,u),\hfill & x\in \Omega ,u=0,\hfill & x\in \partial \Omega ,\end{array}\right.$$ where $\Omega$is a bounded domain in ${ℝ}^n$, $1<p<+\infty$, admits two positive solutions $u_0$, $u_1$ in $W_0^{1,p}\left(\Omega \right)$ such that $0<u_0\le u_1$ in $\Omega$, while $u_0$ is a local minimizer of the associated Euler-Lagrange functional.

The existence of two continuous solutions for a nonlinear singular elliptic equation with natural growth in the gradient is proved for the Dirichlet problem in the unit ball centered at the origin. The first continuous solution is positive and maximal; it is obtained via the regularization method. The second continuous solution is zero at the origin, and follows by considering the corresponding radial ODE and by sub-sup solutions method.

We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $-\Delta u=f_{\lambda }(x,u)$, $u\in H_0^1\left(\Omega \right)$, where $\Omega$ is a bounded domain in ${ℝ}^N$, $N\ge 3$ and $\lambda >0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $\lambda a\left(x\right)u^q+b\left(x\right)u^p$, where $0\le q<1<p\le 2^{*}-1$. The coefficient $a\left(x\right)$ is assumed to be nonnegative but $b\left(x\right)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this...