In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$.

In this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$

We prove the uniqueness of a solution for a problem whose simplest model is

with k ≥ 1, 0 f ∈ L∞(Ω) and Ω is a bounded domain of ℝN, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λuq with 0 < q < 1 and λ > 0.

In this paper, we classify all solutions of

$$\left\{ \begin{align} & -\Delta u=0\,\,\,\,\,\,\,\,\,\text{in}\,\,\,\mathbb{R}_{+}^{2}, \\ & \frac{\partial u}{\partial t}=-c{{\left| x \right|}^{\beta }}{{e}^{u}}\,\,\,\text{on}\,\,\partial \mathbb{R}_{+}^{2}\backslash \left\{ 0 \right\}, \\ \end{align} \right.$$

with the finite conditions

$${{\int }_{\partial \mathbb{R}_{+}^{2}}}|x{{|}^{\beta }}{{e}^{u}}\,ds\,<\,C,\,\,\,\,\frac{\sup }{\mathbb{R}_{+}^{2}}\,u(x)\,<\,C.$$

Here $c$ is a positive number and $\beta \,>\,-1$.

We consider a quasilinear elliptic problem of the form

where λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

In this paper, we prove the existence, uniqueness and multiplicity of positive solutions of a nonlinear perturbed fourth-order problem related to the $Q$ curvature.

We consider the prescribed boundary mean curvature problem in ${{\mathbb{B}}^{N}}$ with the Euclidean metric

$$\{_{\frac{\partial u}{\partial v}+\frac{N-2}{2}u=\frac{N-2}{2}\tilde{K}\left( x \right){{u}^{{{2}^{\#-1}}}}\,\,\,\,\,\,\text{on}{{\mathbb{S}}^{N-1}},}^{-\Delta u=0,\,\,\,\,\,\,u>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in}{{\mathbb{B}}^{N}},}$$

where $\tilde{K}\left( x \right)$ is positive and rotationally symmetric on ${{\mathbb{S}}^{N-1}},{{2}^{\#}}=\frac{2\left( N-1 \right)}{N-2}$. We show that if $\tilde{K}\left( x \right)$ has a local maximum point, then this problem has infinitely many positive solutions that are not rotationally symmetric on ${{\mathbb{S}}^{N-1}}$.

We analyze the regularity of standing wave solutions to nonlinear Schrödinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in Hölder spaces.

We consider a class of critical quasilinear problems

where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain and 1 < p < N, a < N/p, a ≤ b < a + 1, λ is a positive parameter, 0 ≤ μ < ≡ ((N − p)/p − a)p, q = q*(a, b) ≡ Np/[N − pd] and d ≡ a+1 − b. Infinitely many small solutions are obtained by using a version of the symmetric Mountain Pass Theorem and a variant of the concentration-compactness principle. We deal with a problem that extends some results involving singularities not only in the nonlinearities but also in the operator.

In this paper, we show that the semi-linear elliptic systems of the form (0.1) possess at least one non-trivial solution pair (u, v) ∈ H01(Ω) × H01(Ω), where Ω is a smooth bounded domain in ℝN, λ and μ are non-negative numbers, f(x, t) and g(x, t) are continuous functions on Ω × ℝ and asymptotically linear at infinity.

We study the existence of solutions for a class of nonuniformly degenerate elliptic systems in N, N ≥ 3, of the form where hi ∈ L1loc(N), hi(x) ≧ γ0|x|α with α ∈ (0, 2) and γ0 > 0, i = 1, 2. The proofs rely essentially on a variant of the Mountain pass theorem (D. M. Duc, Nonlinear singular elliptic equations, J. Lond. Math. Soc. 40(2) (1989), 420–440) combined with the Caffarelli–Kohn–Nirenberg inequality (First order interpolation inequalities with weights, Composito Math. 53 (1984), 259–275).

Using variational arguments we study the non-existence and multiplicity of non-negative solutions for a class equations of the form where Ω is a bounded domain in N, N ≧ 3, f is a sign-changing Carathéodory function on Ω × [0, +∞) and λ is a positive parameter.

In this paper we study homogenisation problems for Sobolev trace embedding H1(Ω) ↪ Lq(∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, and we consider H1 and Lq spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of a homogenised limit problem, and the best trace constant converges to a homogenised best trace constant. Our results are in fact more general; we can also consider general operators of the form aɛ(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) ↪ Lq(∂Ω).

The aim of this article is to prove a symmetry result for several overdetermined boundary value problems. For the two first problems, our method combines the maximum principle with the monotonicity of the mean curvature. For the others, we use essentially the compatibility condition of the Neumann problem.

In this paper we study the best constant of the Sobolev trace embedding ${{H}^{1}}(\Omega )\,\to \,{{L}^{2}}(\partial \Omega ),$ where $\Omega$ is a bounded smooth domain in ${{\mathbb{R}}^{N}}.$ We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.

In this paper, we show that the semilinear elliptic systems of the form(0.1) possess at least one positive solution pair (u, v) ∈ H 1 0(Ω) × H 1 0(Ω), where Ω is a smooth bounded domain in , f(x,t) and g(x, t) are continuous functions on and asymptotically linear at infinity.

This paper deals with the lack of compactness in the nonlinear elliptic problem $-\Delta u+u=|u|^{p-2}u$ in $\Omega,\ u>0$ in $\Omega,\ u=0$ on $\partial \Omega$, when $\Omega$ is un unbounded domain in $\mathbb{R}^n$ and $2<p<2n/(n-2)$.

In particular, a domain $\widetilde\Omega$ is provided where non-converging Palais–Smale sequences exist at every energy level. Nevertheless, it is proved that the problem has infinitely many solutions on $\widetilde\Omega$.

We study the boundary value problem $\begin{array}{rcl} {\rm div}(|\n u|^{m-2}\n u) + u^av^b &=& 0\quad \mbox{ in } {\Omega}, \vspace{\jot}\\ {\rm div}(|\n v|^{m-2}\n v) + u^cv^d &=& 0 \quad \mbox{ in } {\Omega}, \vspace{\jot}\\ u =v &= & 0 \quad \mbox{ on } {\partial}{\Omega},\vspace{\jot}\\ \end{array}$ where ${\Omega}\subset\mathbb{R}^n$ ($n\ge2$) is a bounded connected smooth domain, and the exponents $m>1$ and $a,b,c,d\ge0$ are non-negative numbers. Under appropriate conditions on the exponents $m$, $a$, $b$, $c$ and $d$, a variety of results on a priori estimates and existence of positive solutions has been established.

Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space $W_{1, p}(D; \rho)$, where the weight function $\rho = \rho(z; \partial D)$ is the distance of the variable point $z$ from the boundary $\partial D$ raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved.

By using Morse theory, we study the existence and multiplicity of nontrivial solutions for a class of p-sublinear p-Laplacian elliptic equations.