A variational principle for problems with a hint of convexity

TL;DR

This paper introduces a new variational principle that extends the ability to solve boundary value problems with a variational structure, especially those beyond traditional compactness constraints, including super-critical elliptic problems.

Contribution

The paper presents a novel variational principle that broadens the scope of solvable boundary value problems beyond weakly compact structures, addressing super-critical elliptic cases.

Findings

Successfully applied to super-critical semilinear elliptic problems

Provides a new framework for boundary value problems with non-compact variational structures

Extends the class of problems solvable via variational methods

Abstract

A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.

Equations

$$\text{Find }{u}_0\in K\text{ such that }D\Psi (u_0)=D\Phi (u_0).$$

$$\displaystyle\Psi_{K}(u)=\left\{\begin{array}[]{cc}\Psi(u),&\quad u\in K,\\ +\infty,&\quad u\not\in K.\end{array}\right.$$

$$I(u):={\Psi }_K(u)-\Phi (u).$$

$${\Psi }_K(v)-{\Psi }_K(u)\ge ⟨D\Phi (u),v-u⟩,\forall v\in V.$$

$$D\Psi (u_0)=D\Phi (u_0).$$

$$\left\{\begin{array}[]{ll}-\Delta u=|u|^{p-2}u+\mu|u|^{q-2}u,&x\in\Omega\\ u=0,&x\in\partial\Omega\end{array}\right.$$

$$I(u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^{2}dx-\frac{1}{p}{\int }_{\Omega }|u{|}^{p}dx-\frac{1}{q}{\int }_{\Omega }|u{|}^{q}dx.$$

$$K(r)=\Big{\{}u\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega);\,\|u\|_{H^{2}(\Omega)}\leq r\Big{\}}.$$

$$\Psi (u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^{2}dx,\Phi (u)=\frac{1}{p}{\int }_{\Omega }|u{|}^{p}dx+{\int }_{\Omega }|u{|}^{q}dx,$$

$$I(u_0)=\underset{u\in K(r)}{\min }I(u).$$

$$\Vert v_0{\Vert }_{H^2(\Omega )}$$

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta u=|u|^{p-2}u+f(x),&\quad x\in\Omega,\\ u=0,&\quad x\in\partial\Omega.\end{array}\right.$$

$$I(u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^{2}dx-\frac{1}{p}{\int }_{\Omega }|u{|}^{p}dx-{\int }_{\Omega }fudx.$$

$$\Psi (u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^{2}dx,\Phi (u)=\frac{1}{p}{\int }_{\Omega }|u{|}^{p}dx+{\int }_{\Omega }fudx,$$

$$K:=K(r)=\big{\{}u\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega);\,\|u\|_{H^{2}(\Omega)}\leq r\big{\}},$$

$$I(u_0)=\underset{u\in K(r)}{\min }I(u).$$

$$\left\{\begin{array}[]{ll}-\Delta u+u=a(x)|u|^{p-2}u,&x\in B_{1}\\ u>0,&x\in B_{1},\\ \frac{\partial u}{\partial\nu}=0,&x\in\partial B_{1},\end{array}\right.$$

$$\Psi (u)={\int }_{\Omega }\frac{|\nabla u{|}^2+u^2}{2}dx,\Phi (u)=\frac{1}{p}{\int }_{\Omega }a(x)|u{|}^{p}dx,$$

$$K=\big{\{}u\in V:u(r)\geq 0,u(r)\leq u(s),\forall r,s\in[0,1],r\leq s\big{\}}.$$

$$\partial\Psi(u)=\big{\{}u^{*}\in V^{*};\langle u^{*},v-u\rangle+\Psi(u)\leq\Psi(v)\text{ for all }v\in V\big{\}}$$

$${\Psi }^{\ast }(u^{\ast })=\sup \{⟨u^{\ast },u⟩-\Psi (u);u\in V\}.$$

$$\Psi (u)+{\Psi }^{\ast }(u^{\ast })=⟨u,u^{\ast }⟩⟺u^{\ast }\in \partial \Psi (u)⟺u\in \partial {\Psi }^{\ast }(u^{\ast }).$$

$${\Psi }_K(v)-{\Psi }_K(u_0)\ge ⟨D\Phi (u_0),v-u_0⟩,\forall v\in V.$$

$$\Psi (v_0)-\Psi (u_0)\ge ⟨D\Psi (v_0),v_0-u_0⟩.$$

$$\Psi (u_0)-\Psi (v_0)\ge ⟨D\Psi (v_0),u_0-v_0⟩.$$

$$\Psi (v_0)-\Psi (u_0)=⟨D\Psi (v_0),v_0-u_0⟩.$$

$$D\Psi (u_0)=D\Psi (v_0)=D\Phi (u_0).$$

$$\Psi (v_0)+{\Psi }^{\ast }(w^{\ast })=⟨w^{\ast },v_0⟩.$$

$$⟨w^{\ast },u_0⟩-\Psi (u_0)=⟨w^{\ast },v_0⟩-\Psi (v_0)={\Psi }^{\ast }(w^{\ast }),$$

Full text

A variational principle for problems with a hint of convexity

111The author is pleased to acknowledge the support of the National Sciences and Engineering Research Council of Canada.

Abbas Moameni   

*School of Mathematics and Statistics

Carleton University,* *Ottawa, ON, Canada

momeni@math.carleton.ca

Abstract

A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.

2010 Mathematics Subject Classification: 35J15, 58E30.

Key words: Variational principles, supercritical Elliptic problems.

1 Introduction

Let $V$ be a real Banach space and $V^{*}$ its topological dual and let $\langle.,.\rangle$ be the pairing between $V$ and $V^{*}.$ Let $\Psi:V\rightarrow\mathbb{R}\cup\{+\infty\}$ be a proper convex and lower semi continuous function and let $K$ be a convex and weakly closed subset of $V.$ Assume that $\Psi$ is Gâteaux differentiable on $K$ and denote by $D\Psi$ the Gâteaux derivative of $\Psi.$ Let $\Phi\in C^{1}(V,\R)$ and consider the following problem,

[TABLE]

The restriction of $\Psi$ to $K$ is denoted by $\Psi_{K}$ and defined by

[TABLE]

To find a solution for (1), we shall consider the critical points of the functional $I:V\rightarrow\mathbb{R}\cup\{+\infty\}$ defined by

[TABLE]

According to Szulkin [16] we have the following definition for critical points of $I$ (see also the appendix).

Definition 1.1**.**

A point $u\in V$ is said to be a critical point of $I$ if $I(u)\in\R$ and it satisfies the inequality

[TABLE]

Note that a function $u$ satisfying (3) is indeed a solution of the inclusion $D\Phi(u)\in\partial\Psi_{K}(u).$ Therefore, it is not necessarily a solution of (1) unless $D=V.$ There is a well developed theory to find critical points of functionals of the form $I$. We refer the interested reader to [16, 13]. Here is our main result in this paper.

Theorem 1.2** (Variational Principle).**

Let $\Psi:V\rightarrow\mathbb{R}\cup\{+\infty\}$ be a proper convex and lower semi continuous function and let $K$ be a convex and weakly closed subset of $V.$ Assume that $\Psi$ is Gâteaux differentiable on $K$ and $\Phi\in C^{1}(V,\R)$. If the following two assertions hold:

1. (i)

The functional $I:V\rightarrow\mathbb{R}\cup\{+\infty\}$ defined by $I(u)=\Psi_{K}(u)-\Phi(u)$ has a critical point $u_{0}\in V,$ and; 2. (ii)

there exists $v_{0}\in K$ such that $D\Psi(v_{0})=D\Phi(u_{0}).$

Then $u_{0}\in K$ is a solution of (1), that is,

[TABLE]

The above theorem has many interesting applications in partial differential equations . We shall briefly recall some of them and refer the interested reader to [12] where some more general versions of Theorem 1.2 are established and several applications in the fixed point theory and PDEs are provided. It is also worth noting that Theorem 1.2 extends some of variational principles established by the author in [10, 11].

We shall now proceed with some applications.

1.1 A concave-convex nonlinearity

We consider the problem

[TABLE]

where $\Omega\subset{}^{n}$ is a bounded domain with $C^{1}$-boundary and $1<q\leq 2<p.$ This problem was studied by Ambrosetti and etc. in [1] and Bartsch and Willem in [3]. Our plan is to show that for positive $\mu$ and $p$ bigger that the critical exponent $2^{*}=2n/(n-2),$ problem (4) has a strong solution in $H^{2}(\Omega).$

Let $V=H^{2}(\Omega)\cap H_{0}^{1}(\Omega),$ and let $I:V\to\R$ be the Euler-Lagrange functional corresponding to (7),

[TABLE]

For $r>0$, define the convex set $K(r)$ by

[TABLE]

We have the following result

Theorem 1.3**.**

Assume that $1<q<2<p<p^{*}$ where $p^{*}=(2n-4)/(n-4)$ for $n>4$ and $p^{*}=\infty$ for $n\leq 4.$ Then there exists $\mu^{*}>0$ such that for each $\mu\in(0,\mu^{*})$ problem (4) has a non-trivial solution. Indeed, for each $\mu\in(0,\mu^{*}),$ there exist positive numbers $r_{1},r_{2}\in\R$ with $r_{1}<r_{2}$ such that for each $r\in[r_{1},r_{2}]$ the problem (4) has a solution $u\in K(r)$ with $I(u)<0.$

Proof.

We apply Theorem 1.2, where

[TABLE]

and $K:=K(r)$ for some $r>0$ to be determined. Note that the Sobolev space $H^{2}(\Omega)$ is compactly embedded in $L^{t}(\Omega)$ for $t<t^{*}$ where $t^{*}=2n/(n-4)$ for $n>4,$ and $t^{*}=+\infty$ for $n\leq 4.$ It then follows that the function $\Phi$ is continuously differentiable for $p<p^{*}.$ By standard methods, there exists $u_{0}\in K(r)$ such that

[TABLE]

Since $1<q<2<p$ and $\mu>0,$ it is easily seen that $I(u_{0})<0$ and therefore $u_{0}\not\equiv 0$ is a critical point of $I$ restricted to $K(r).$ To verify condition $(ii)$ in Theorem 1.2, we show that there exists $v_{0}\in K(r)$ such that $-\Delta v_{0}=|u_{0}|^{p-2}u_{0}+\mu|u_{0}|^{q-2}u_{0}.$ The existence of such $v_{0}$ follows by standard arguments. We show that $v_{0}\in K(r)$ for $r$ small. It follows from the Elliptic regularity theory (see Theorem 8.12 in [8]) that

[TABLE]

where $C$ is a constant depending on $\Omega.$ Since $2(q-1)<2(p-1)<t^{*},$ we obtain that

[TABLE]

where $C_{1}$ is a constant in terms of $p,q$ and $\Omega.$ Choose $\mu^{*}>0$ small enough such that for each $\mu\in(0,\mu^{*}),$ there exist positive numbers $r_{1},r_{2}\in\R$ with $r_{1}<r_{2}$ such that $C_{1}(r^{p-1}+\mu r^{q-1})\leq r$ for all $r\in[r_{1},r_{2}].$ It then follows that $v_{0}\in K(r)$ provided $\mu\in(0,\mu^{*})$ and $r\in[r_{1},r_{2}].$

∎

1.2 Non-homogeneous semilinear Elliptic equations

Here we shall consider the problem

[TABLE]

where $\Omega$ is on open bounded domain in n with $C^{1}$-boundary. Problem (7) was treated in [2, 15] for $p$ less than the critical exponent $2^{*}.$ As an application of Theorem 1.2 together with Elliptic regularity theory we shall show that problem (7) has a solution for $p$ beyond the critical Sobolev exponent. In this case, the standard variational methods fail to work. Note that our approach can be applied to more general nonlinearities (see [12]). We have the following theorem.

Theorem 1.4**.**

Let $2<p<p^{*}$ where $p^{*}=(2n-4)/(n-4)$ for $n>4$ and $p^{*}=\infty$ for $n\leq 4.$ There exists $\lambda>0$ such that for $\|f\|_{L^{2}(\Omega)}<\lambda,$ problem (7) has a solution $u\in H^{2}(\Omega).$

Proof.

Let $V=H^{2}(\Omega)\cap H_{0}^{1}(\Omega),$ and let $I:V\to\R$ be the Euler-Lagrange functional corresponding to (7),

[TABLE]

We apply Theorem 1.2, where

[TABLE]

and

[TABLE]

for some $r>0$ to be determined. By standard methods, there exists $u_{0}\in K(r)$ such that

[TABLE]

To verify condition $(ii)$ in Theorem 1.2, one needs to show that there exists $v_{0}\in K(r)$ such that $-\Delta v_{0}=|u_{0}|^{p-2}u_{0}+f(x).$ Existence of $v_{0}\in H^{2}(\Omega)$ is standard. The fact that $v_{0}\in K(r)$ for $\|f\|_{L^{2}(\Omega)}$ small, follows by the Elliptic regularity theory and the argument made in the proof of Theorem 1.3. ∎

1.3 Super critical Neumann problems

We shall consider the existence of positive solutions of the Neumann problem

[TABLE]

where $B_{1}$ is the unit ball centered at the origin in $\mathbb{R}^{N}$, $N\geq 3,$ $p>2.$ and $a$ is a radial function, i.e., $a(x)=a(r)$ where $r=|x|.$

Theorem 1.5**.**

Assume that $a\in L^{\infty}(0,1)$ is increasing, not constant and $a(r)>0$ a.e. in $[0,1]$. Then problem (8) admits at least one radially increasing positive solution.

Sketch of the proof. Let $V=L^{p}(\Omega)\cap H_{r}^{1}(\Omega),$ where $H^{1}_{r}$ is the set of radial functions in $H^{1}(\Omega).$ We apply Theorem 1.2, where

[TABLE]

and

[TABLE]

It can be easily deduced that that $V\cap K$ is continuously embedded in $L^{\infty}(\Omega)$ from which one can apply Theorem 3.3 to show that $I=\Psi-\Phi$ restricted to $K$ has a critical point $u_{0}\in K$ of mountain pass type (See [5] for a detailed argument). It is also established in [5] that there exists $v_{0}\in K$ satisfying $-\Delta v_{0}+v_{0}=a(|x|)|u_{0}|^{p-2}u_{0}.$ Thus, by Theorem 1.2, $u_{0}$ is a non-negative and nontrivial solution of (8). It also follows from the maximum principle that $u_{0}$ is indeed positive. $\square$

We remark that finding radially increasing solutions of problems of type (8) has been the subject of many studies in recent years starting the works of [4, 9, 14].

2 Proof of the variatinal principle.

In this section we shall prove Theorem 1.2. We first recall some important definitions and results from convex analysis.

Let $V$ be a real Banach space and $V^{*}$ its topological dual and let $\langle.,.\rangle$ be the pairing between $V$ and $V^{*}.$ Let $\Psi:V\rightarrow\mathbb{R}\cup\{\infty\}$ be a proper convex function. The subdifferential $\partial\Psi$ of $\Psi$ is defined to be the following set-valued operator: if $u\in Dom(\Psi)=\{v\in V;\,\Psi(v)<\infty\},$ set

[TABLE]

and if $u\not\in Dom(\Psi),$ set $\partial\Psi(u)=\varnothing.$ If $\Psi$ is Gâteaux differentiable at $u,$ denote by $D\Psi(u)$ the derivative of $\Psi$ at $u.$ In this case $\partial\Psi(u)=\{D\Psi(u)\}.$

The Fenchel dual of an arbitrary function $\Psi$ is denoted by $\Psi^{*},$ that is function on $V^{*}$ and is defined by

[TABLE]

Clearly $\Psi^{*}:V^{*}\rightarrow\mathbb{R}\cup\{+\infty\}$ is convex and weakly lower semi-continuous. The following standard result is crucial in the subsequent analysis (see [7] for a proof).

Proposition 2.1**.**

Let $\Psi:V\rightarrow\mathbb{R}\cup\{+\infty\}$ be convex and lower-semi continuous. then $\Psi^{**}=\Psi$ and the following holds:

[TABLE]

Proof of Theorem 1.2. Since $u_{0}$ is a critical point of $I(u)=\Psi_{K}(u)-\Phi(u),$ it follows from Definition 1.1 that

[TABLE]

It follows from $(i)$ and $(ii)$ in the theorem that $u_{0},v_{0}\in K$ and $D\Psi(v_{0})=D\Phi(u_{0})$. Thus, it follows from inequality (9) with $v=v_{0}$ that

[TABLE]

Since $\Psi$ is Gâteaux differentiable at $v_{0}\in K,$ it follows that $\partial\Psi(v_{0})=\{D\Psi(v_{0})\}$ which together with the convexity of $\Psi$ one obtains that

[TABLE]

It follows from (10) and (11) that

[TABLE]

We now claim that $D\Psi(v_{0})=D\Psi(u_{0})$ from which the desired result follows,

[TABLE]

Proof of the claim: Let $w^{*}=D\Psi(v_{0}).$ Since $\Psi$ is convex and lower semi continuous it follows from Proposition 2.1 that

[TABLE]

It now follows from (12) and (13) that

[TABLE]

from which one obtains

[TABLE]

This indeed implies that $w^{*}\in\partial\Psi(u_{0})$ by virtue of Proposition 2.1. Since $\Psi$ is Gâteaux differentiable at $u_{0}$ we have that $\partial\Psi(u_{0})=\{D\Psi(u_{0})\}$. Therefore,

[TABLE]

as claimed. $\square$

3 Appendix

We shall now recall some notations and results for the minimax principles of lower semi-continuous functions used throughout the paper.

Definition 3.1**.**

Let $V$ be a real Banach space, $\Phi\in C^{1}(V,\mathbb{R})$ and $\Psi:V\rightarrow(-\infty,+\infty]$ be proper (i.e. $Dom(\Psi)\neq\emptyset$), convex and lower semi-continuous. A point $u\in V$ is said to be a critical point of

[TABLE]

if $u\in Dom(\Psi)$ and if it satisfies the inequality

[TABLE]

Definition 3.2**.**

We say that $I$ satisfies the Palais-Smale compactness condition (PS) if every sequence $\{u_{n}\}$ such that $I(u_{n})\rightarrow c\in\mathbb{R},$ and

[TABLE]

where $\epsilon_{n}\rightarrow 0$, then $\{u_{n}\}$ possesses a convergent subsequence.

The following is proved in [16].

Theorem 3.3**.**

(Mountain Pass Theorem). Suppose that $I:V\rightarrow(-\infty,+\infty]$ is of the form (14) and satisfies the Palais-Smale condition and the Mountain Pass Geometry (MPG):

$I(0)=0$. and there exists $e\in V$ such that $I(e)\leq 0$. 2. 2.

there exists some $\rho$ such that $0<\rho<\|e\|$ and for every $u\in V$ with $\|u\|=\rho$ one has $I(u)>0$.

Then $I$ has a critical value $c\geq\rho$ which is characterized by

[TABLE]

where $\Gamma=\{g\in C([0,1],V):g(0)=0,g(1)=e\}.$