Diffeomorphism-invariant properties for quasi-linear elliptic operators

For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and non-degenerate coerciveness.Comment: 16 page

Critical Points for Some Functionals of the Calculus of Variations

In this paper we prove the existence of critical points of non differentiable functionals of the kind J(v)=\frac_\Omega A(x,v)\nabla v\cdot\nabla v-\frac1{p+1}\int_\Omega (v^+)^{p+1}, where $1 2$, $p> 1$ if $N\leq 2$ and $v^+$ stands for the positive part of the function $v$. The coefficient $A(x,s)=(a_{ij}(x,s))$ is a Carathéodory matrix derivable with respect to the variable $s$. Even if both $A(x,s)$ and $A'_s(x,s)$ are uniformly bounded by positive constants, the functional $J$ fails to be differentiable on $H^1_0(\Omega)$. Indeed, $J$ is only derivable along directions of $H^1_0(\Omega)\cap L^{\infty}(\Omega)$ so that the classical critical point theory cannot be applied. We will prove the existence of a critical point of $J$ by assuming that there exist positive continuous functions $\alpha(s)$, $\beta(s)$ and a positive constants $\alpha_0$ and $M$ satisfying $\alpha_0|\xi|^2\leq \alpha(s)|\xi|^2 \leq A(x,s)\xi\cdot \xi$, $A(x,0)\leq M$, $|A'_s(x,s)|\leq \beta(s)$, with $\beta(s)$ in $L^1(\mathbb R)$

Quasi--Linear equations on R^N: Perturbation Results

In this paper we prove existence of nontrivial solutions for the quasi-linear elliptic problem {div((I + epsilonA(x, u))delu) + u + epsilonH(x, u,delu) = \u\(p-1), in R-N, u is an element of H-1(R-N) boolean AND W-2,W-q (R-N), q > N where 1 2 and the operator -div((I + epsilonA(x, u))delu) +epsilonH(x, u, delu) is a perturbation of the Laplacian. We use a perturbation method recently developed in [1], [2], [3] and we get results both in the variational and in the non-variational framework

Critical Points for Non Differentiable Functionals

In this paper we deal with the existence and multiplicity of critical points for non differentiable integral functionals defined in the Sobolev space W1,p(Ω) (p > 1) by: 0 where Ω is a bounded open set of RN, with N ≥ 3 and p ≤ N. Under natural assump- tions F turns out to be not Frech ́et differentiable on W1,p(Ω), thus classical critical point theory cannot be applied. The existence of a critical point of F has been proved in [1] by means of a suitable extension of the Ambrosetti-Rabinowitz minimax result. Here we get existence and multiplicity of critical points of F applying a generalization of a symmetric version of the Mountain-Pass theorem proved in [10]. We will follow the same procedure of [7] where the quasilinear case has been treated

Local estimates and global existence for nonlinear parabolic equations with absorbing lower order terms

We obtain existence results for some strongly nonlinear Cauchy problems posed in R^N and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudo-monotone operator of Leray-Lions type acting on L^p (0, T ; W^1,p(R^N )), they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence results

Critical points of non-regular integral functionals

We prove the existence of a bounded positive critical point for a class of functionals such as J(v)=\frac12\io [a(x)+b(x)|v|^{\gamma}]|\nabla v|^{2}-\io |v|^{p} for $\Omega$ a bounded open set in $\R^{N}$, $N>2$, $\gamma+20$, $\gamma\neq 1$ and $a(x),\,b(x)$ measurable function satisfying $0<\alpha\leq a(x)\leq \beta$, $0\leq b(x)\leq\beta$ almost everywhere in $\Omega$